Deficient & Abundant Number Fun

[Qfwfq]

can't we find the number of divisors by finding the number of combinations of the prime factors?

Certainly, but it's because of how this works that I find 319 and 383 surprising. But it seems Donk meant number of divisors not counting the number itself. Counting them as 320 and 384 it's already a bit better.

 

[math]320=2^7\cdot5^1[/math]

 

[math]384=2^8\cdot3^1[/math]

 

Does that make any kind of sense?

It makes abundant sense indeed.

 

The way to determine how many factors a given number has from its prime factorization depends solely on the count of prime factors and their multiplicities but not on the value of each prime. So, two numbers with the same number of distinct prime factors and also the same multiplicities necessarily have the same number of divisors. Let's consider:

 

[math]2^a\cdot3^b\cdot5^c\cdot7^d[/math]

 

[math]23^c\cdot37^a\cdot73^d\cdot1019^b[/math]

 

[math]47^d\cdot257^c\cdot947^a\cdot1571^b[/math]

 

They have the same number of divisors, which is [imath](a+1)(b+1)(c+1)(d+1)[/imath] including 1 and the number itself. Those are just three of many with the same count, for a given choice of multiplicities. There can also be different multiplicities with the same product. Consider [imath]c^{\prime}=(c+1)(d+1)-1[/imath] and you can obtain numbers with three distinct prime factors and the same number of divisors as the above examples.

 

[math]17640=2^3\cdot3^2\cdot5^1\cdot7^2[/math] ought to have [math]96=4\cdot3\cdot2\cdot3[/math] divisors.

 

[math]225000=2^3\cdot3^2\cdot5^5[/math] ought to have the same number of divisors too: [math]96=4\cdot3\cdot6[/math].

 

Now in Donk's cases there are [math]2\cdot7[/math] and [math]2\cdot8[/math] divisors of 320 and of 384 (which are 319 and 383 when the number itself isn't counted) and this makes for many ways of getting those same two numbers of divisors. It's a bit trickier to count the number of products having the same result, than just the number of divisors. For each divisor, if the result isn't prime, there also choices for the remaining factors, so perhaps it is quicker to split the prime factorization into all non-equivalent subsets; for powers of a single prime this means summing the incremented exponents and seeing how many sums add to the same total. In general it's a bit more tricky so I'll have to think about it.

 

I know, I know, it's hard to follow cuz it's a bit of a recursivey thingy.

[Donk]

The first fruits of my labours (or to be more precise, my trusty little desktop comp).

 

Since +/-20 wasn't giving many hits, I've increased the range to +/-50 which gives 372 hits in the first 10 million. Compared to most numbers, they still qualify as "nearly perfect". Quasi-perfect? To save unnecessary scrolling, I've put zero and -1 at the bottom. Not that perfect numbers and powers of 2 aren't interesting, but they're known values. No surprises there.

 

The list runs:

abundant/deficient : number : prime factors : total of all factors. I've taken out the full factors list, just leaving the primes. I've attached the full list, including composite factors, in csv format.

2 : 20 : 2^2,5 : (22)
2 : 104 : 2^3,13 : (106)
2 : 464 : 2^4,29 : (466)
2 : 650 : 2,5^2,13 : (652)
2 : 1952 : 2^5,61 : (1954)
2 : 130304 : 2^8,509 : (130306)
2 : 522752 : 2^9,1021 : (522754)
2 : 8382464 : 2^11,4093 : (8382466)
-2 : 10 : 2,5 : (8)
-2 : 136 : 2^3,17 : (134)
-2 : 32896 : 2^7,257 : (32894)
3 : 18 : 2,3^2 : (21)
4 : 12 : 2^2,3 : (16)
4 : 70 : 2,5,7 : (74)
4 : 88 : 2^3,11 : (92)
4 : 1888 : 2^5,59 : (1892)
4 : 4030 : 2,5,13,31 : (4034)
4 : 5830 : 2,5,11,53 : (5834)
4 : 32128 : 2^7,251 : (32132)
4 : 521728 : 2^9,1019 : (521732)
4 : 1848964 : 2^2,13,31^2,37 : (1848968)
4 : 8378368 : 2^11,4091 : (8378372)
-4 : 14 : 2,7 : (10)
-4 : 44 : 2^2,11 : (40)
-4 : 110 : 2,5,11 : (106)
-4 : 152 : 2^3,19 : (148)
-4 : 884 : 2^2,13,17 : (880)
-4 : 2144 : 2^5,67 : (2140)
-4 : 8384 : 2^6,131 : (8380)
-4 : 18632 : 2^3,17,137 : (18628)
-4 : 116624 : 2^4,37,197 : (116620)
-4 : 8394752 : 2^11,4099 : (8394748)
-5 : 9 : 3^2 : (4)
6 : 8925 : 3,5^2,7,17 : (8931)
6 : 32445 : 3^2,5,7,103 : (32451)
6 : 442365 : 3,5,7,11,383 : (442371)
-6 : 15 : 3,5 : (9)
-6 : 52 : 2^2,13 : (46)
-6 : 315 : 3^2,5,7 : (309)
-6 : 592 : 2^4,37 : (586)
-6 : 1155 : 3,5,7,11 : (1149)
-6 : 2102272 : 2^10,2053 : (2102266)
7 : 196 : 2^2,7,7 : (203)
-7 : 50 : 2,5,5 : (43)
8 : 56 : 2^3,7 : (64)
8 : 368 : 2^4,23 : (376)
8 : 836 : 2^2,11,19 : (844)
8 : 11096 : 2^3,19,73 : (11104)
8 : 17816 : 2^3,17,131 : (17824)
8 : 45356 : 2^2,17,23,29 : (45364)
8 : 77744 : 2^4,43,113 : (77752)
8 : 91388 : 2^2,11,31,67 : (91396)
8 : 128768 : 2^8,503 : (128776)
8 : 254012 : 2^2,11,23,251 : (254020)
8 : 388076 : 2^2,13,17,439 : (388084)
8 : 2087936 : 2^10,2039 : (2087944)
8 : 2291936 : 2^5,67,1069 : (2291944)
-8 : 22 : 2,11 : (14)
-8 : 130 : 2,5,13 : (122)
-8 : 184 : 2^3,23 : (176)
-8 : 1012 : 2^2,11,23 : (1004)
-8 : 2272 : 2^5,71 : (2264)
-8 : 18904 : 2^3,17,139 : (18896)
-8 : 33664 : 2^7,263 : (33656)
-8 : 70564 : 2^2,13,23,59 : (70556)
-8 : 85936 : 2^4,41,131 : (85928)
-8 : 100804 : 2^2,11,29,79 : (100796)
-8 : 391612 : 2^2,13,17,443 : (391604)
-8 : 527872 : 2^9,1031 : (527864)
-8 : 1090912 : 2^5,73,467 : (1090904)
10 : 40 : 2^3,5 : (50)
10 : 1696 : 2^5,53 : (1706)
10 : 518656 : 2^9,1013 : (518666)
-10 : 21 : 3,7 : (11)
-10 : 26 : 2,13 : (16)
-10 : 68 : 2^2,17 : (58)
-10 : 656 : 2^4,41 : (646)
-10 : 2336 : 2^5,73 : (2326)
-10 : 8768 : 2^6,137 : (8758)
-10 : 133376 : 2^8,521 : (133366)
-10 : 528896 : 2^9,1033 : (528886)
-11 : 244036 : 2^2,13,13,19,19 : (244025)
-12 : 45 : 3^2,5 : (33)
-12 : 76 : 2^2,19 : (64)
-12 : 688 : 2^4,43 : (676)
-12 : 8896 : 2^6,139 : (8884)
-12 : 133888 : 2^8,523 : (133876)
14 : 272 : 2^4,17 : (286)
14 : 7232 : 2^6,113 : (7246)
14 : 30848 : 2^7,241 : (30862)
14 : 516608 : 2^9,1009 : (516622)
-14 : 27 : 3^3 : (13)
-14 : 34 : 2,17 : (20)
-14 : 232 : 2^3,29 : (218)
-14 : 34432 : 2^7,269 : (34418)
16 : 550 : 2,5^2,11 : (566)
16 : 748 : 2^2,11,17 : (764)
16 : 1504 : 2^5,47 : (1520)
16 : 7192 : 2^3,29,31 : (7208)
16 : 7912 : 2^3,23,43 : (7928)
16 : 10792 : 2^3,19,71 : (10808)
16 : 17272 : 2^3,17,127 : (17288)
16 : 30592 : 2^7,239 : (30608)
16 : 1713592 : 2^3,23,67,139 : (1713608)
16 : 4526272 : 2^6,197,359 : (4526288)
16 : 8353792 : 2^11,4079 : (8353808)
16 : 9928792 : 2^3,19,83,787 : (9928808)
-16 : 38 : 2,19 : (22)
-16 : 92 : 2^2,23 : (76)
-16 : 170 : 2,5,17 : (154)
-16 : 248 : 2^3,31 : (232)
-16 : 752 : 2^4,47 : (736)
-16 : 988 : 2^2,13,19 : (972)
-16 : 2528 : 2^5,79 : (2512)
-16 : 8648 : 2^3,23,47 : (8632)
-16 : 12008 : 2^3,19,79 : (11992)
-16 : 34688 : 2^7,271 : (34672)
-16 : 63248 : 2^4,59,67 : (63232)
-16 : 117808 : 2^4,37,199 : (117792)
-16 : 526688 : 2^5,109,151 : (526672)
-16 : 531968 : 2^9,1039 : (531952)
-16 : 820808 : 2^3,37,47,59 : (820792)
-16 : 1292768 : 2^5,71,569 : (1292752)
-16 : 1495688 : 2^3,31,37,163 : (1495672)
-16 : 2095208 : 2^3,23,59,193 : (2095192)
-16 : 2112512 : 2^10,2063 : (2112496)
-16 : 3477608 : 2^3,19,137,167 : (3477592)
-16 : 4495808 : 2^6,199,353 : (4495792)
-16 : 8419328 : 2^11,4111 : (8419312)
17 : 100 : 2^2,5^2 : (117)
18 : 208 : 2^4,13 : (226)
18 : 6976 : 2^6,109 : (6994)
18 : 8415 : 3^2,5,11,17 : (8433)
18 : 31815 : 3^2,5,7,101 : (31833)
18 : 351351 : 3^3,7,11,13^2 : (351369)
18 : 2077696 : 2^10,2029 : (2077714)
-18 : 33 : 3,11 : (15)
-18 : 105 : 3,5,7 : (87)
-18 : 33705 : 3^2,5,7,107 : (33687)
19 : 36 : 2^2,3^2 : (55)
-19 : 25 : 5^2 : (6)
-19 : 2312 : 2^3,17^2 : (2293)
20 : 176 : 2^4,11 : (196)
20 : 1376 : 2^5,43 : (1396)
20 : 3230 : 2,5,17,19 : (3250)
20 : 3770 : 2,5,13,29 : (3790)
20 : 6848 : 2^6,107 : (6868)
20 : 114256 : 2^4,37,193 : (114276)
20 : 125696 : 2^8,491 : (125716)
20 : 544310 : 2,5,13,53,79 : (544330)
20 : 561824 : 2^5,97,181 : (561844)
20 : 740870 : 2,5,13,41,139 : (740890)
20 : 2075648 : 2^10,2027 : (2075668)
20 : 4199030 : 2,5,11,59,647 : (4199050)
20 : 4607296 : 2^6,193,373 : (4607316)
20 : 8436950 : 2,5^2,19,83,107 : (8436970)
-20 : 46 : 2,23 : (26)
-20 : 154 : 2,7,11 : (134)
-20 : 190 : 2,5,19 : (170)
-20 : 2656 : 2^5,83 : (2636)
-20 : 6490 : 2,5,11,59 : (6470)
-20 : 44650 : 2,5^2,19,47 : (44630)
-20 : 318250 : 2,5^3,19,67 : (318230)
-20 : 1360810 : 2,5,11,89,139 : (1360790)
-20 : 1503370 : 2,5,11,79,173 : (1503350)
-20 : 1788490 : 2,5,11,71,229 : (1788470)
-20 : 3214090 : 2,5,11,61,479 : (3214070)
22 : 1312 : 2^5,41 : (1334)
22 : 29824 : 2^7,233 : (29846)
22 : 8341504 : 2^11,4073 : (8341526)
-22 : 35 : 5,7 : (13)
-22 : 39 : 3,13 : (17)
-22 : 63 : 3^2,7 : (41)
-22 : 116 : 2^2,29 : (94)
-22 : 296 : 2^3,37 : (274)
-22 : 848 : 2^4,53 : (826)
-22 : 9536 : 2^6,149 : (9514)
-22 : 35456 : 2^7,277 : (35434)
-22 : 2118656 : 2^10,2069 : (2118634)
24 : 112 : 2^4,7 : (136)
24 : 6592 : 2^6,103 : (6616)
24 : 124672 : 2^8,487 : (124696)
-24 : 124 : 2^2,31 : (100)
-24 : 9664 : 2^6,151 : (9640)
-25 : 98 : 2,7^2 : (73)
26 : 80 : 2^4,5 : (106)
26 : 1184 : 2^5,37 : (1210)
26 : 6464 : 2^6,101 : (6490)
26 : 29312 : 2^7,229 : (29338)
26 : 78975 : 3^5,5^2,13 : (79001)
26 : 510464 : 2^9,997 : (510490)
26 : 557192 : 2^3,17^2,241 : (557218)
-26 : 58 : 2,29 : (32)
-26 : 75 : 3,5^2 : (49)
-26 : 328 : 2^3,41 : (302)
-26 : 850 : 2,5^2,17 : (824)
-26 : 1210 : 2,5,11^2 : (1184)
-26 : 2848 : 2^5,89 : (2822)
-26 : 35968 : 2^7,281 : (35942)
-26 : 537088 : 2^9,1049 : (537062)
28 : 48 : 2^4,3 : (76)
28 : 2002 : 2,7,11,13 : (2030)
28 : 5170 : 2,5,11,47 : (5198)
28 : 29056 : 2^7,227 : (29084)
28 : 133042 : 2,7,13,17,43 : (133070)
-28 : 62 : 2,31 : (34)
-28 : 182 : 2,7,13 : (154)
-28 : 230 : 2,5,23 : (202)
-28 : 344 : 2^3,43 : (316)
-28 : 944 : 2^4,59 : (916)
-28 : 6710 : 2,5,11,61 : (6682)
-28 : 20264 : 2^3,17,149 : (20236)
-28 : 36224 : 2^7,283 : (36196)
-28 : 538112 : 2^9,1051 : (538084)
-28 : 2085710 : 2,5,11,67,283 : (2085682)
30 : 945 : 3^3,5,7 : (975)
30 : 6208 : 2^6,97 : (6238)
30 : 7425 : 3^3,5,5,11 : (7455)
30 : 15028 : 2^2,13,17,17 : (15058)
30 : 437745 : 3,5,7,11,379 : (437775)
30 : 2065408 : 2^10,2017 : (2065438)
30 : 3428368 : 2^4,47^2,97 : (3428398)
-30 : 51 : 3,17 : (21)
-30 : 135 : 3^3,5 : (105)
-30 : 148 : 2^2,37 : (118)
-30 : 976 : 2^4,61 : (946)
-30 : 10048 : 2^6,157 : (10018)
-30 : 34335 : 3^2,5,7,109 : (34305)
-30 : 138496 : 2^8,541 : (138466)
-30 : 449295 : 3,5,7,11,389 : (449265)
31 : 15376 : 2^4,31,31 : (15407)
32 : 572 : 2^2,11,13 : (604)
32 : 992 : 2^5,31 : (1024)
32 : 7544 : 2^3,23,41 : (7576)
32 : 10184 : 2^3,19,67 : (10216)
32 : 28544 : 2^7,223 : (28576)
32 : 83312 : 2^4,41,127 : (83344)
32 : 113072 : 2^4,37,191 : (113104)
32 : 122624 : 2^8,479 : (122656)
32 : 382772 : 2^2,13,17,433 : (382804)
32 : 507392 : 2^9,991 : (507424)
32 : 537248 : 2^5,103,163 : (537280)
32 : 698528 : 2^5,83,263 : (698560)
32 : 791264 : 2^5,79,313 : (791296)
32 : 1081568 : 2^5,73,463 : (1081600)
32 : 1279136 : 2^5,71,563 : (1279168)
32 : 2154584 : 2^3,29,37,251 : (2154616)
32 : 2279072 : 2^5,67,1063 : (2279104)
32 : 5029184 : 2^6,179,439 : (5029216)
-32 : 250 : 2,5^3 : (218)
-32 : 376 : 2^3,47 : (344)
-32 : 1276 : 2^2,11,29 : (1244)
-32 : 12616 : 2^3,19,83 : (12584)
-32 : 20536 : 2^3,17,151 : (20504)
-32 : 396916 : 2^2,13,17,449 : (396884)
-32 : 801376 : 2^5,79,317 : (801344)
-32 : 1297312 : 2^5,71,571 : (1297280)
-32 : 8452096 : 2^11,4127 : (8452064)
34 : 928 : 2^5,29 : (962)
-34 : 57 : 3,19 : (23)
-34 : 74 : 2,37 : (40)
-34 : 164 : 2^2,41 : (130)
-34 : 3104 : 2^5,97 : (3070)
-34 : 2130944 : 2^10,2081 : (2130910)
-34 : 8456192 : 2^11,4129 : (8456158)
36 : 2205 : 3^2,5,7^2 : (2241)
36 : 2059264 : 2^10,2011 : (2059300)
-36 : 172 : 2^2,43 : (136)
-36 : 1072 : 2^4,67 : (1036)
-36 : 10432 : 2^6,163 : (10396)
-36 : 140032 : 2^8,547 : (139996)
-36 : 2132992 : 2^10,2083 : (2132956)
-37 : 484 : 2^2,11^2 : (447)
38 : 5696 : 2^6,89 : (5734)
38 : 8308736 : 2^11,4057 : (8308774)
-38 : 55 : 5,11 : (17)
-38 : 82 : 2,41 : (44)
-38 : 424 : 2^3,53 : (386)
-38 : 3232 : 2^5,101 : (3194)
-38 : 37504 : 2^7,293 : (37466)
-38 : 543232 : 2^9,1061 : (543194)
-38 : 8464384 : 2^11,4133 : (8464346)
39 : 162 : 2,3^4 : (201)
40 : 736 : 2^5,23 : (776)
40 : 381004 : 2^2,13,17,431 : (381044)
40 : 503296 : 2^9,983 : (503336)
40 : 2274784 : 2^5,67,1061 : (2274824)
-40 : 86 : 2,43 : (46)
-40 : 188 : 2^2,47 : (148)
-40 : 290 : 2,5,29 : (250)
-40 : 950 : 2,5^2,19 : (910)
-40 : 1136 : 2^4,71 : (1096)
-40 : 1196 : 2^2,13,23 : (1156)
-40 : 1364 : 2^2,11,31 : (1324)
-40 : 3296 : 2^5,103 : (3256)
-40 : 10688 : 2^6,167 : (10648)
-40 : 260084 : 2^2,11,23,257 : (260044)
-40 : 544256 : 2^9,1063 : (544216)
-40 : 2137088 : 2^10,2087 : (2137048)
41 : 1352 : 2^3,13^2 : (1393)
-41 : 49 : 7^2 : (8)
-41 : 81 : 3^4 : (40)
42 : 9555 : 3,5,7^2,13 : (9597)
42 : 30555 : 3^2,5,7,97 : (30597)
-42 : 69 : 3,23 : (27)
-42 : 99 : 3^2,11 : (57)
-42 : 165 : 3,5,11 : (123)
-42 : 1168 : 2^4,73 : (1126)
-42 : 1365 : 3,5,7,13 : (1323)
-42 : 2139136 : 2^10,2089 : (2139094)
44 : 350 : 2,5^2,7 : (394)
44 : 608 : 2^5,19 : (652)
44 : 4730 : 2,5,11,43 : (4774)
44 : 5312 : 2^6,83 : (5356)
44 : 15368 : 2^3,17,113 : (15412)
44 : 27008 : 2^7,211 : (27052)
44 : 119552 : 2^8,467 : (119596)
44 : 2051072 : 2^10,2003 : (2051116)
44 : 8296448 : 2^11,4051 : (8296492)
-44 : 94 : 2,47 : (50)
-44 : 238 : 2,7,17 : (194)
-44 : 310 : 2,5,31 : (266)
-44 : 472 : 2^3,59 : (428)
-44 : 3424 : 2^5,107 : (3380)
-44 : 3910 : 2,5,17,23 : (3866)
-44 : 4810 : 2,5,13,37 : (4766)
-44 : 21352 : 2^3,17,157 : (21308)
-44 : 2765092 : 2^2,11^2,29,197 : (2765048)
-44 : 8476672 : 2^11,4139 : (8476628)
46 : 490 : 2,5,7^2 : (536)
46 : 544 : 2^5,17 : (590)
46 : 500224 : 2^9,977 : (500270)
46 : 8292352 : 2^11,4049 : (8292398)
-46 : 65 : 5,13 : (19)
-46 : 212 : 2^2,53 : (166)
-46 : 488 : 2^3,61 : (442)
-46 : 3488 : 2^5,109 : (3442)
-46 : 11072 : 2^6,173 : (11026)
-46 : 142592 : 2^8,557 : (142546)
-46 : 547328 : 2^9,1069 : (547282)
-47 : 225 : 3^2,5^2 : (178)
48 : 60 : 2^2,3,5 : (108)
48 : 5056 : 2^6,79 : (5104)
48 : 118528 : 2^8,463 : (118576)
48 : 2046976 : 2^10,1999 : (2047024)
-48 : 1264 : 2^4,79 : (1216)
0 : 6 : 2,3 : (6)
0 : 28 : 2^2,7 : (28)
0 : 496 : 2^4,31 : (496)
0 : 8128 : 2^6,127 : (8128)
-1 : 4 : 2^2 : (3)
-1 : 8 : 2^3 : (7)
-1 : 16 : 2^4 : (15)
-1 : 32 : 2^5 : (31)
-1 : 64 : 2^6 : (63)
-1 : 128 : 2^7 : (127)
-1 : 256 : 2^8 : (255)
-1 : 512 : 2^9 : (511)
-1 : 1024 : 2^10 : (1023)
-1 : 2048 : 2^11 : (2047)
-1 : 4096 : 2^12 : (4095)
-1 : 8192 : 2^13 : (8191)
-1 : 16384 : 2^14 : (16383)
-1 : 32768 : 2^15 : (32767)
-1 : 65536 : 2^16 : (65535)
-1 : 131072 : 2^17 : (131071)
-1 : 262144 : 2^18 : (262143)
-1 : 524288 : 2^19 : (524287)
-1 : 1048576 : 2^20 : (1048575)
-1 : 2097152 : 2^21 : (2097151)
-1 : 4194304 : 2^22 : (4194303)
-1 : 8388608 : 2^23 : (8388607)

A few patterns catch the eye.

 

There are a lot of powers of 2 there, but 3's a bit quiet. And 2 and 3 in combination is even rarer: 6, 12, 18, 36, 48, 60, 162... then nothing as far as the eye can see (up to 10 million, anyway). Low numbers would tend to have low abundance values, so the sub-200s don't really count. We know about 6*p, of course, but does this mean that all numbers of the form 6n, where n is not prime, are abundant/deficient by more than 50?

 

And talking of 3s - most "quasi-perfects" with a factor of 3 are abundant/deficient by a number which also divides by 3. Exceptions are 9, 21, 27, 36, 39, 57, 63, 75, 81, 225... and 78975!

 

Then there are the sequences. -16 has a sequence of 6 numbers of the form 2^n*p; as does -26. -36 has a sequence of 5 of the form 2^(2n)*p, and -38 has 6 of the form 2^(2n+1)*p

 

Will those sequences continue? Watch out for the next exciting episode! :hihi: ;)

[Turtle]

The first fruits of my labours (or to be more precise, my trusty little desktop comp).

 

Since +/-20 wasn't giving many hits, I've increased the range to +/-50 which gives 372 hits in the first 10 million. Compared to most numbers, they still qualify as "nearly perfect". Quasi-perfect?

 

excellent, and no. :eek: :D that is, no we can't say "quasi-perfect" because that term is already used to designate specifically a number abundant-by-1.

Quasiperfect number - Wikipedia, the free encyclopedia

Originally Posted by wican pestidia

No quasiperfect numbers have been found so far, but if a quasiperfect number exists, it must be an odd square number greater than 10^35 and have at least seven distinct prime factors.

 

To save unnecessary scrolling, I've put zero and -1 at the bottom. Not that perfect numbers and powers of 2 aren't interesting, but they're known values. No surprises there. ...

no surprises for powers of 2 in our interval (can we prove that all powers of 2 are deficient-by-1? i have only seen it asserted as the case without proof. :rose:), however it is not known if a number exists that is deficient-by-1 and not a power of 2. we can call that an unknown value for fun. :lol:

 

The list runs:

abundant/deficient : number : prime factors : total of all factors. I've taken out the full factors list, just leaving the primes. I've attached the full list, including composite factors, in csv format.

8 : 56 : 2^3,7 : (64)
8 : 368 : 2^4,23 : (376)
8 : 836 : 2^2,11,19 : (844)
8 : 11096 : 2^3,19,73 : (11104)
8 : 17816 : 2^3,17,131 : (17824)
8 : 45356 : 2^2,17,23,29 : (45364)
8 : 77744 : 2^4,43,113 : (77752)
8 : 91388 : 2^2,11,31,67 : (91396)
8 : 128768 : 2^8,503 : (128776)
8 : 254012 : 2^2,11,23,251 : (254020)
8 : 388076 : 2^2,13,17,439 : (388084)
8 : 2087936 : 2^10,2039 : (2087944)
8 : 2291936 : 2^5,67,1069 : (2291944)

 

i redacted the list to just the A8's as i have been searching them all this time & i'm just now @ 11740000+ with a programmed stop @12000000. i found no more and my list matches yours perfectly. :doh: :hyper: :steering:

 

A few patterns catch the eye.

 

There are a lot of powers of 2 there, but 3's a bit quiet. And 2 and 3 in combination is even rarer: 6, 12, 18, 36, 48, 60, 162... then nothing as far as the eye can see (up to 10 million, anyway). Low numbers would tend to have low abundance values, so the sub-200s don't really count. We know about 6*p, of course, but does this mean that all numbers of the form 6n, where n is not prime, are abundant/deficient by more than 50?

i gotta cogitate on your eye candy, , but 6*6=36 and 36 is abundant-by-19. i also recall that a multiple of either a perfect or an abundant number is an abundant number.

 

And talking of 3s - most "quasi-perfects" with a factor of 3 are abundant/deficient by a number which also divides by 3. Exceptions are 9, 21, 27, 36, 39, 57, 63, 75, 81, 225... and 78975!

 

Then there are the sequences. -16 has a sequence of 6 numbers of the form 2^n*p; as does -26. -36 has a sequence of 5 of the form 2^(2n)*p, and -38 has 6 of the form 2^(2n+1)*p

 

Will those sequences continue? Watch out for the next exciting episode! :hyper:

 

processing....

[Donk]

excellent, and no. :steering: that is, no we can't say "quasi-perfect" because that term is already used to designate specifically a number abundant-by-1.

In that case, I propose we call them Jan's numbers. Nearly perfect, but with enough imperfection to be interesting, surprising, occasionally infuriating and all-round bloody wonderful!

 

And yes, I'm biased

[Don Blazys]

Quoting Donk:

2 : 20 : 2^2,5 : (22)

2 : 104 : 2^3,13 : (106)

2 : 464 : 2^4,29 : (466)

2 : 650 : 2,5^2,13 : (652)

2 : 1952 : 2^5,61 : (1954)

2 : 130304 : 2^8,509 : (130306)

2 : 522752 : 2^9,1021 : (522754)

2 : 8382464 : 2^11,4093 : (8382466)

 

20+3=23 (prime)

104+3=107 (prime)

464+3=467 (prime)

650+3=653 (prime)

1952+3=1955 (composite...divisible by 5)

130304+3=130307 (prime)

522752+3=522755 (composite...divisible by 5)

8382464+3=8382467 (prime)

 

If this were to be continued,

would all resulting composites

be divisible by 5 ?

 

Don.

[Turtle]

The first fruits of my labours (or to be more precise, my trusty little desktop comp).

...

A few patterns catch the eye. ...

 

eye-catchers missing from jan's corral:

 

interval <=10000000

no numbers abundant-by-5

no numbers abundant-by-9 or deficient-by-9

no numbers abundant-by-11

no numbers abundant-by-13 or deficient-by-13

no numbers abundant-by-15 or deficient-by-15

no numbers deficient-by-17

no numbers abundant-by-23 or deficient-by-23

no numbers abundant-by-25

no numbers abundant-by-27 or deficient-by-27

no numbers abundant-by-29 or deficient-by-29

no numbers deficient-by-31

no numbers abundant-by-33 or deficient-by-33

no numbers abundant-by-35 or deficient-by-35

no numbers abundant-by-37

no numbers deficient-by-39

no numbers abundant-by-41

no numbers abundant-by-43 or deficient-by-43

no numbers abundant-by-45 or deficient-by-45

no numbers abundant-by-47

no numbers abundant-by-49 or deficient-by-49

no numbers abundant-by-50 or deficient-by-50

 

that's all i got...erhm...haven't got. . . . . . :hyper:

[Donk]

The programme has ticked its way past 500 million, and it's finding very few new Jan's numbers. 452 found so far, of which 443 were in the first 250 million, and only 9 in the second! Should I increase the limits from +/- 50 to +/- 200? Would that tell us anything interesting, or just confuse matters?

 

I'm away for the weekend, so I'll be shutting it down tomorrow morning, restarting on Monday evening.

2 : 20 : 2^2,5:(22)
2 : 104 : 2^3,13:(106)
2 : 464 : 2^4,29:(466)
2 : 650 : 2,5^2,13:(652)
2 : 1952 : 2^5,61:(1954)
2 : 130304 : 2^8,509:(130306)
2 : 522752 : 2^9,1021:(522754)
2 : 8382464 : 2^11,4093:(8382466)
2 : 134193152 : 2^13,16381:(134193154)
-2 : 10 : 2,5:(8)
-2 : 136 : 2^3,17:(134)
-2 : 32896 : 2^7,257:(32894)
3 : 18 : 2,3^2:(21)
4 : 12 : 2^2,3:(16)
4 : 70 : 2,5,7:(74)
4 : 88 : 2^3,11:(92)
4 : 1888 : 2^5,59:(1892)
4 : 4030 : 2,5,13,31:(4034)
4 : 5830 : 2,5,11,53:(5834)
4 : 32128 : 2^7,251:(32132)
4 : 521728 : 2^9,1019:(521732)
4 : 1848964 : 2^2,13,31^2,37:(1848968)
4 : 8378368 : 2^11,4091:(8378372)
-4 : 14 : 2,7:(10)
-4 : 44 : 2^2,11:(40)
-4 : 110 : 2,5,11:(106)
-4 : 152 : 2^3,19:(148)
-4 : 884 : 2^2,13,17:(880)
-4 : 2144 : 2^5,67:(2140)
-4 : 8384 : 2^6,131:(8380)
-4 : 18632 : 2^3,17,137:(18628)
-4 : 116624 : 2^4,37,197:(116620)
-4 : 8394752 : 2^11,4099:(8394748)
-4 : 15370304 : 2^6,137,1753:(15370300)
-4 : 73995392 : 2^7,293,1973:(73995388)
-5 : 9 : 3^2:(4)
6 : 8925 : 3,5^2,7,17:(8931)
6 : 32445 : 3^2,5,7,103:(32451)
6 : 442365 : 3,5,7,11,383:(442371)
-6 : 15 : 3,5:(9)
-6 : 52 : 2^2,13:(46)
-6 : 315 : 3^2,5,7:(309)
-6 : 592 : 2^4,37:(586)
-6 : 1155 : 3,5,7,11:(1149)
-6 : 2102272 : 2^10,2053:(2102266)
7 : 196 : 2^2,7^2:(203)
-7 : 50 : 2,5^2:(43)
8 : 56 : 2^3,7:(64)
8 : 368 : 2^4,23:(376)
8 : 836 : 2^2,11,19:(844)
8 : 11096 : 2^3,19,73:(11104)
8 : 17816 : 2^3,17,131:(17824)
8 : 45356 : 2^2,17,23,29:(45364)
8 : 77744 : 2^4,43,113:(77752)
8 : 91388 : 2^2,11,31,67:(91396)
8 : 128768 : 2^8,503:(128776)
8 : 254012 : 2^2,11,23,251:(254020)
8 : 388076 : 2^2,13,17,439:(388084)
8 : 2087936 : 2^10,2039:(2087944)
8 : 2291936 : 2^5,67,1069:(2291944)
8 : 13174976 : 2^6,139,1481:(13174984)
8 : 29465852 : 2^2,13,23,71,347:(29465860)
8 : 35021696 : 2^7,419,653:(35021704)
8 : 45335936 : 2^7,337,1051:(45335944)
8 : 120888092 : 2^2,13,23,61,1657:(120888100)
8 : 260378492 : 2^2,11,23,457,563:(260378500)
8 : 381236216 : 2^3,19^2,101,1307:(381236224)
-8 : 22 : 2,11:(14)
-8 : 130 : 2,5,13:(122)
-8 : 184 : 2^3,23:(176)
-8 : 1012 : 2^2,11,23:(1004)
-8 : 2272 : 2^5,71:(2264)
-8 : 18904 : 2^3,17,139:(18896)
-8 : 33664 : 2^7,263:(33656)
-8 : 70564 : 2^2,13,23,59:(70556)
-8 : 85936 : 2^4,41,131:(85928)
-8 : 100804 : 2^2,11,29,79:(100796)
-8 : 391612 : 2^2,13,17,443:(391604)
-8 : 527872 : 2^9,1031:(527864)
-8 : 1090912 : 2^5,73,467:(1090904)
-8 : 17619844 : 2^2,11,37,79,137:(17619836)
10 : 40 : 2^3,5:(50)
10 : 1696 : 2^5,53:(1706)
10 : 518656 : 2^9,1013:(518666)
-10 : 21 : 3,7:(11)
-10 : 26 : 2,13:(16)
-10 : 68 : 2^2,17:(58)
-10 : 656 : 2^4,41:(646)
-10 : 2336 : 2^5,73:(2326)
-10 : 8768 : 2^6,137:(8758)
-10 : 133376 : 2^8,521:(133366)
-10 : 528896 : 2^9,1033:(528886)
-11 : 244036 : 2^2,13^2,19^2:(244025)
-12 : 45 : 3^2,5:(33)
-12 : 76 : 2^2,19:(64)
-12 : 688 : 2^4,43:(676)
-12 : 8896 : 2^6,139:(8884)
-12 : 133888 : 2^8,523:(133876)
14 : 272 : 2^4,17:(286)
14 : 7232 : 2^6,113:(7246)
14 : 30848 : 2^7,241:(30862)
14 : 516608 : 2^9,1009:(516622)
14 : 134094848 : 2^13,16369:(134094862)
-14 : 27 : 3^3:(13)
-14 : 34 : 2,17:(20)
-14 : 232 : 2^3,29:(218)
-14 : 34432 : 2^7,269:(34418)
16 : 550 : 2,5^2,11:(566)
16 : 748 : 2^2,11,17:(764)
16 : 1504 : 2^5,47:(1520)
16 : 7192 : 2^3,29,31:(7208)
16 : 7912 : 2^3,23,43:(7928)
16 : 10792 : 2^3,19,71:(10808)
16 : 17272 : 2^3,17,127:(17288)
16 : 30592 : 2^7,239:(30608)
16 : 1713592 : 2^3,23,67,139:(1713608)
16 : 4526272 : 2^6,197,359:(4526288)
16 : 8353792 : 2^11,4079:(8353808)
16 : 9928792 : 2^3,19,83,787:(9928808)
16 : 11547352 : 2^3,17,197,431:(11547368)
16 : 17999992 : 2^3,19,79,1499:(18000008)
16 : 89283592 : 2^3,17,139,4723:(89283608)
16 : 173482552 : 2^3,17,137,9311:(173482568)
16 : 361702144 : 2^8,677,2087:(361702160)
-16 : 38 : 2,19:(22)
-16 : 92 : 2^2,23:(76)
-16 : 170 : 2,5,17:(154)
-16 : 248 : 2^3,31:(232)
-16 : 752 : 2^4,47:(736)
-16 : 988 : 2^2,13,19:(972)
-16 : 2528 : 2^5,79:(2512)
-16 : 8648 : 2^3,23,47:(8632)
-16 : 12008 : 2^3,19,79:(11992)
-16 : 34688 : 2^7,271:(34672)
-16 : 63248 : 2^4,59,67:(63232)
-16 : 117808 : 2^4,37,199:(117792)
-16 : 526688 : 2^5,109,151:(526672)
-16 : 531968 : 2^9,1039:(531952)
-16 : 820808 : 2^3,37,47,59:(820792)
-16 : 1292768 : 2^5,71,569:(1292752)
-16 : 1495688 : 2^3,31,37,163:(1495672)
-16 : 2095208 : 2^3,23,59,193:(2095192)
-16 : 2112512 : 2^10,2063:(2112496)
-16 : 3477608 : 2^3,19,137,167:(3477592)
-16 : 4495808 : 2^6,199,353:(4495792)
-16 : 8419328 : 2^11,4111:(8419312)
-16 : 12026888 : 2^3,17,191,463:(12026872)
-16 : 13192768 : 2^6,139,1483:(13192752)
-16 : 16102808 : 2^3,17,167,709:(16102792)
-16 : 26347688 : 2^3,17,151,1283:(26347672)
-16 : 29322008 : 2^3,17,149,1447:(29321992)
-16 : 33653888 : 2^7,467,563:(33653872)
-16 : 169371008 : 2^7,269,4919:(169370992)
-16 : 173631608 : 2^3,17,137,9319:(173631592)
-16 : 293947648 : 2^8,787,1459:(293947632)
17 : 100 : 2^2,5^2:(117)
18 : 208 : 2^4,13:(226)
18 : 6976 : 2^6,109:(6994)
18 : 8415 : 3^2,5,11,17:(8433)
18 : 31815 : 3^2,5,7,101:(31833)
18 : 351351 : 3^3,7,11,13^2:(351369)
18 : 2077696 : 2^10,2029:(2077714)
18 : 20487159 : 3^2,7,11,17,37,47:(20487177)
18 : 159030135 : 3^5,5,11,73,163:(159030153)
-18 : 33 : 3,11:(15)
-18 : 105 : 3,5,7:(87)
-18 : 33705 : 3^2,5,7,107:(33687)
-18 : 33624064 : 2^12,8209:(33624046)
19 : 36 : 2^2,3^2:(55)
-19 : 25 : 5^2:(6)
-19 : 2312 : 2^3,17^2:(2293)
20 : 176 : 2^4,11:(196)
20 : 1376 : 2^5,43:(1396)
20 : 3230 : 2,5,17,19:(3250)
20 : 3770 : 2,5,13,29:(3790)
20 : 6848 : 2^6,107:(6868)
20 : 114256 : 2^4,37,193:(114276)
20 : 125696 : 2^8,491:(125716)
20 : 544310 : 2,5,13,53,79:(544330)
20 : 561824 : 2^5,97,181:(561844)
20 : 740870 : 2,5,13,41,139:(740890)
20 : 2075648 : 2^10,2027:(2075668)
20 : 4199030 : 2,5,11,59,647:(4199050)
20 : 4607296 : 2^6,193,373:(4607316)
20 : 8436950 : 2,5^2,19,83,107:(8436970)
20 : 33468416 : 2^12,8171:(33468436)
20 : 134045696 : 2^13,16363:(134045716)
20 : 199272950 : 2,5^2,19,47,4463:(199272970)
-20 : 46 : 2,23:(26)
-20 : 154 : 2,7,11:(134)
-20 : 190 : 2,5,19:(170)
-20 : 2656 : 2^5,83:(2636)
-20 : 6490 : 2,5,11,59:(6470)
-20 : 44650 : 2,5^2,19,47:(44630)
-20 : 318250 : 2,5^3,19,67:(318230)
-20 : 1360810 : 2,5,11,89,139:(1360790)
-20 : 1503370 : 2,5,11,79,173:(1503350)
-20 : 1788490 : 2,5,11,71,229:(1788470)
-20 : 3214090 : 2,5,11,61,479:(3214070)
-20 : 103712410 : 2,5,17,29,109,193:(103712390)
22 : 1312 : 2^5,41:(1334)
22 : 29824 : 2^7,233:(29846)
22 : 8341504 : 2^11,4073:(8341526)
22 : 134029312 : 2^13,16361:(134029334)
-22 : 35 : 5,7:(13)
-22 : 39 : 3,13:(17)
-22 : 63 : 3^2,7:(41)
-22 : 116 : 2^2,29:(94)
-22 : 296 : 2^3,37:(274)
-22 : 848 : 2^4,53:(826)
-22 : 9536 : 2^6,149:(9514)
-22 : 35456 : 2^7,277:(35434)
-22 : 2118656 : 2^10,2069:(2118634)
24 : 112 : 2^4,7:(136)
24 : 6592 : 2^6,103:(6616)
24 : 124672 : 2^8,487:(124696)
24 : 33452032 : 2^12,8167:(33452056)
-24 : 124 : 2^2,31:(100)
-24 : 9664 : 2^6,151:(9640)
-25 : 98 : 2,7^2:(73)
26 : 80 : 2^4,5:(106)
26 : 1184 : 2^5,37:(1210)
26 : 6464 : 2^6,101:(6490)
26 : 29312 : 2^7,229:(29338)
26 : 78975 : 3^5,5^2,13:(79001)
26 : 510464 : 2^9,997:(510490)
26 : 557192 : 2^3,17^2,241:(557218)
-26 : 58 : 2,29:(32)
-26 : 75 : 3,5^2:(49)
-26 : 328 : 2^3,41:(302)
-26 : 850 : 2,5^2,17:(824)
-26 : 1210 : 2,5,11^2:(1184)
-26 : 2848 : 2^5,89:(2822)
-26 : 35968 : 2^7,281:(35942)
-26 : 537088 : 2^9,1049:(537062)
28 : 48 : 2^4,3:(76)
28 : 2002 : 2,7,11,13:(2030)
28 : 5170 : 2,5,11,47:(5198)
28 : 29056 : 2^7,227:(29084)
28 : 133042 : 2,7,13,17,43:(133070)
28 : 114203776 : 2^7,277,3221:(114203804)
-28 : 62 : 2,31:(34)
-28 : 182 : 2,7,13:(154)
-28 : 230 : 2,5,23:(202)
-28 : 344 : 2^3,43:(316)
-28 : 944 : 2^4,59:(916)
-28 : 6710 : 2,5,11,61:(6682)
-28 : 20264 : 2^3,17,149:(20236)
-28 : 36224 : 2^7,283:(36196)
-28 : 538112 : 2^9,1051:(538084)
-28 : 2085710 : 2,5,11,67,283:(2085682)
-28 : 14503550 : 2,5^2,17,113,151:(14503522)
-28 : 33665024 : 2^12,8219:(33664996)
-28 : 55328384 : 2^7,313,1381:(55328356)
-28 : 134438912 : 2^13,16411:(134438884)
30 : 945 : 3^3,5,7:(975)
30 : 6208 : 2^6,97:(6238)
30 : 7425 : 3^3,5^2,11:(7455)
30 : 15028 : 2^2,13,17^2:(15058)
30 : 437745 : 3,5,7,11,379:(437775)
30 : 2065408 : 2^10,2017:(2065438)
30 : 3428368 : 2^4,47^2,97:(3428398)
30 : 20355825 : 3,5^2,7^2,29,191:(20355855)
30 : 33427456 : 2^12,8161:(33427486)
30 : 78524145 : 3^2,5,7,109,2287:(78524175)
-30 : 51 : 3,17:(21)
-30 : 135 : 3^3,5:(105)
-30 : 148 : 2^2,37:(118)
-30 : 976 : 2^4,61:(946)
-30 : 10048 : 2^6,157:(10018)
-30 : 34335 : 3^2,5,7,109:(34305)
-30 : 138496 : 2^8,541:(138466)
-30 : 449295 : 3,5,7,11,389:(449265)
-30 : 33673216 : 2^12,8221:(33673186)
31 : 15376 : 2^4,31^2:(15407)
32 : 572 : 2^2,11,13:(604)
32 : 992 : 2^5,31:(1024)
32 : 7544 : 2^3,23,41:(7576)
32 : 10184 : 2^3,19,67:(10216)
32 : 28544 : 2^7,223:(28576)
32 : 83312 : 2^4,41,127:(83344)
32 : 113072 : 2^4,37,191:(113104)
32 : 122624 : 2^8,479:(122656)
32 : 382772 : 2^2,13,17,433:(382804)
32 : 507392 : 2^9,991:(507424)
32 : 537248 : 2^5,103,163:(537280)
32 : 698528 : 2^5,83,263:(698560)
32 : 791264 : 2^5,79,313:(791296)
32 : 1081568 : 2^5,73,463:(1081600)
32 : 1279136 : 2^5,71,563:(1279168)
32 : 2154584 : 2^3,29,37,251:(2154616)
32 : 2279072 : 2^5,67,1063:(2279104)
32 : 5029184 : 2^6,179,439:(5029216)
32 : 15126992 : 2^4,67,103,137:(15127024)
32 : 29581424 : 2^4,47,139,283:(29581456)
32 : 74899952 : 2^4,47,103,967:(74899984)
32 : 89245784 : 2^3,17,139,4721:(89245816)
32 : 95327216 : 2^4,43,127,1091:(95327248)
32 : 307801856 : 2^8,751,1601:(307801888)
-32 : 250 : 2,5^3:(218)
-32 : 376 : 2^3,47:(344)
-32 : 1276 : 2^2,11,29:(1244)
-32 : 12616 : 2^3,19,83:(12584)
-32 : 20536 : 2^3,17,151:(20504)
-32 : 396916 : 2^2,13,17,449:(396884)
-32 : 801376 : 2^5,79,317:(801344)
-32 : 1297312 : 2^5,71,571:(1297280)
-32 : 8452096 : 2^11,4127:(8452064)
-32 : 33721216 : 2^7,463,569:(33721184)
-32 : 40575616 : 2^7,359,883:(40575584)
-32 : 59376256 : 2^7,307,1511:(59376224)
-32 : 89397016 : 2^3,17,139,4729:(89396984)
-32 : 99523456 : 2^7,281,2767:(99523424)
-32 : 101556016 : 2^4,41,149,1039:(101555984)
-32 : 150441856 : 2^7,271,4337:(150441824)
-32 : 173706136 : 2^3,17,137,9323:(173706104)
-32 : 269096704 : 2^8,953,1103:(269096672)
-32 : 283417216 : 2^7,263,8419:(283417184)
-32 : 500101936 : 2^4,59,67,7907:(500101904)
34 : 928 : 2^5,29:(962)
34 : 133931008 : 2^13,16349:(133931042)
-34 : 57 : 3,19:(23)
-34 : 74 : 2,37:(40)
-34 : 164 : 2^2,41:(130)
-34 : 3104 : 2^5,97:(3070)
-34 : 2130944 : 2^10,2081:(2130910)
-34 : 8456192 : 2^11,4129:(8456158)
-34 : 134488064 : 2^13,16417:(134488030)
36 : 2205 : 3^2,5,7^2:(2241)
36 : 2059264 : 2^10,2011:(2059300)
-36 : 172 : 2^2,43:(136)
-36 : 1072 : 2^4,67:(1036)
-36 : 10432 : 2^6,163:(10396)
-36 : 140032 : 2^8,547:(139996)
-36 : 2132992 : 2^10,2083:(2132956)
-37 : 484 : 2^2,11^2:(447)
38 : 5696 : 2^6,89:(5734)
38 : 8308736 : 2^11,4057:(8308774)
-38 : 55 : 5,11:(17)
-38 : 82 : 2,41:(44)
-38 : 424 : 2^3,53:(386)
-38 : 3232 : 2^5,101:(3194)
-38 : 37504 : 2^7,293:(37466)
-38 : 543232 : 2^9,1061:(543194)
-38 : 8464384 : 2^11,4133:(8464346)
-38 : 134520832 : 2^13,16421:(134520794)
39 : 162 : 2,3^4:(201)
40 : 736 : 2^5,23:(776)
40 : 381004 : 2^2,13,17,431:(381044)
40 : 503296 : 2^9,983:(503336)
40 : 2274784 : 2^5,67,1061:(2274824)
-40 : 86 : 2,43:(46)
-40 : 188 : 2^2,47:(148)
-40 : 290 : 2,5,29:(250)
-40 : 950 : 2,5^2,19:(910)
-40 : 1136 : 2^4,71:(1096)
-40 : 1196 : 2^2,13,23:(1156)
-40 : 1364 : 2^2,11,31:(1324)
-40 : 3296 : 2^5,103:(3256)
-40 : 10688 : 2^6,167:(10648)
-40 : 260084 : 2^2,11,23,257:(260044)
-40 : 544256 : 2^9,1063:(544216)
-40 : 2137088 : 2^10,2087:(2137048)
-40 : 33714176 : 2^12,8231:(33714136)
-40 : 35221184 : 2^6,131,4201:(35221144)
41 : 1352 : 2^3,13^2:(1393)
-41 : 49 : 7^2:(8)
-41 : 81 : 3^4:(40)
42 : 9555 : 3,5,7^2,13:(9597)
42 : 30555 : 3^2,5,7,97:(30597)
-42 : 69 : 3,23:(27)
-42 : 99 : 3^2,11:(57)
-42 : 165 : 3,5,11:(123)
-42 : 1168 : 2^4,73:(1126)
-42 : 1365 : 3,5,7,13:(1323)
-42 : 2139136 : 2^10,2089:(2139094)
-42 : 32062485 : 3,5,7,13,83,283:(32062443)
-42 : 33722368 : 2^12,8233:(33722326)
-42 : 132701205 : 3,5,7,13,67,1451:(132701163)
44 : 350 : 2,5^2,7:(394)
44 : 608 : 2^5,19:(652)
44 : 4730 : 2,5,11,43:(4774)
44 : 5312 : 2^6,83:(5356)
44 : 15368 : 2^3,17,113:(15412)
44 : 27008 : 2^7,211:(27052)
44 : 119552 : 2^8,467:(119596)
44 : 2051072 : 2^10,2003:(2051116)
44 : 8296448 : 2^11,4051:(8296492)
44 : 33370112 : 2^12,8147:(33370156)
44 : 133849088 : 2^13,16339:(133849132)
-44 : 94 : 2,47:(50)
-44 : 238 : 2,7,17:(194)
-44 : 310 : 2,5,31:(266)
-44 : 472 : 2^3,59:(428)
-44 : 3424 : 2^5,107:(3380)
-44 : 3910 : 2,5,17,23:(3866)
-44 : 4810 : 2,5,13,37:(4766)
-44 : 21352 : 2^3,17,157:(21308)
-44 : 2765092 : 2^2,11^2,29,197:(2765048)
-44 : 8476672 : 2^11,4139:(8476628)
-44 : 42305350 : 2,5^2,17,71,701:(42305306)
-44 : 134569984 : 2^13,16427:(134569940)
46 : 490 : 2,5,7^2:(536)
46 : 544 : 2^5,17:(590)
46 : 500224 : 2^9,977:(500270)
46 : 8292352 : 2^11,4049:(8292398)
-46 : 65 : 5,13:(19)
-46 : 212 : 2^2,53:(166)
-46 : 488 : 2^3,61:(442)
-46 : 3488 : 2^5,109:(3442)
-46 : 11072 : 2^6,173:(11026)
-46 : 142592 : 2^8,557:(142546)
-46 : 547328 : 2^9,1069:(547282)
-46 : 33738752 : 2^12,8237:(33738706)
-47 : 225 : 3^2,5^2:(178)
48 : 60 : 2^2,3,5:(108)
48 : 5056 : 2^6,79:(5104)
48 : 118528 : 2^8,463:(118576)
48 : 2046976 : 2^10,1999:(2047024)
-48 : 1264 : 2^4,79:(1216)
0 : 6 : 2,3:(6)
0 : 28 : 2^2,7:(28)
0 : 496 : 2^4,31:(496)
0 : 8128 : 2^6,127:(8128)
0 : 33550336 : 2^12,8191:(33550336)
-1 : 4 : 2^2:(3)
-1 : 8 : 2^3:(7)
-1 : 16 : 2^4:(15)
-1 : 32 : 2^5:(31)
-1 : 64 : 2^6:(63)
-1 : 128 : 2^7:(127)
-1 : 256 : 2^8:(255)
-1 : 512 : 2^9:(511)
-1 : 1024 : 2^10:(1023)
-1 : 2048 : 2^11:(2047)
-1 : 4096 : 2^12:(4095)
-1 : 8192 : 2^13:(8191)
-1 : 16384 : 2^14:(16383)
-1 : 32768 : 2^15:(32767)
-1 : 65536 : 2^16:(65535)
-1 : 131072 : 2^17:(131071)
-1 : 262144 : 2^18:(262143)
-1 : 524288 : 2^19:(524287)
-1 : 1048576 : 2^20:(1048575)
-1 : 2097152 : 2^21:(2097151)
-1 : 4194304 : 2^22:(4194303)
-1 : 8388608 : 2^23:(8388607)
-1 : 16777216 : 2^24:(16777215)
-1 : 33554432 : 2^25:(33554431)
-1 : 67108864 : 2^26:(67108863)
-1 : 134217728 : 2^27:(134217727)
-1 : 268435456 : 2^28:(268435455)

[Turtle]

The programme has ticked its way past 500 million, and it's finding very few new Jan's numbers. 452 found so far, of which 443 were in the first 250 million, and only 9 in the second! Should I increase the limits from +/- 50 to +/- 200? Would that tell us anything interesting, or just confuse matters?

 

I'm away for the weekend, so I'll be shutting it down tomorrow morning, restarting on Monday evening.

...

 

outstanding!! looks like none of the missing jan's that i listed showed up. i was thinking maybe to finish your +/- 50 to the 1-3 billion interval that you mentioned & then start again at 1 looking for +/- 51 to 100. this might reduce confusion while giving another comparative tool.

 

we might follow up on your noted forms and generate a list of values from your expressions & then check the output values individualy for their perfection/abundance/deficience.

Then there are the sequences. -16 has a sequence of 6 numbers of the form 2^n*p; as does -26. -36 has a sequence of 5 of the form 2^(2n)*p, and -38 has 6 of the form 2^(2n+1)*p

 

Will those sequences continue? Watch out for the next exciting episode!

 

that's all i got. . . . . .

[Donk]

The good news is that we've reached 991,000,000.

 

BUT, the even better news is that my Lady :doh: is finally back from a loooooong weekend!

 

So I'll be posting the results tomorrow night ;)

[Donk]

Normal programming resumed following a much-needed break

 

We've now trudged our way up to a billion. There are 477 numbers abundant/deficient by less than 50:

-2 : 10 : 2,5 (8)
-2 : 136 : 2^3,17 (134)
-2 : 32896 : 2^7,257 (32894)
2 : 20 : 2^2,5 (22)
2 : 104 : 2^3,13 (106)
2 : 464 : 2^4,29 (466)
2 : 650 : 2,5^2,13 (652)
2 : 1952 : 2^5,61 (1954)
2 : 130304 : 2^8,509 (130306)
2 : 522752 : 2^9,1021 (522754)
2 : 8382464 : 2^11,4093 (8382466)
2 : 134193152 : 2^13,16381 (134193154)
3 : 18 : 2,3^2 (21)
-4 : 14 : 2,7 (10)
-4 : 44 : 2^2,11 (40)
-4 : 110 : 2,5,11 (106)
-4 : 152 : 2^3,19 (148)
-4 : 884 : 2^2,13,17 (880)
-4 : 2144 : 2^5,67 (2140)
-4 : 8384 : 2^6,131 (8380)
-4 : 18632 : 2^3,17,137 (18628)
-4 : 116624 : 2^4,37,197 (116620)
-4 : 8394752 : 2^11,4099 (8394748)
-4 : 15370304 : 2^6,137,1753 (15370300)
-4 : 73995392 : 2^7,293,1973 (73995388)
-4 : 536920064 : 2^14,32771 (536920060)
4 : 12 : 2^2,3 (16)
4 : 70 : 2,5,7 (74)
4 : 88 : 2^3,11 (92)
4 : 1888 : 2^5,59 (1892)
4 : 4030 : 2,5,13,31 (4034)
4 : 5830 : 2,5,11,53 (5834)
4 : 32128 : 2^7,251 (32132)
4 : 521728 : 2^9,1019 (521732)
4 : 1848964 : 2^2,13,31^2,37 (1848968)
4 : 8378368 : 2^11,4091 (8378372)
-5 : 9 : 3^2 (4)
-6 : 15 : 3,5 (9)
-6 : 52 : 2^2,13 (46)
-6 : 315 : 3^2,5,7 (309)
-6 : 592 : 2^4,37 (586)
-6 : 1155 : 3,5,7,11 (1149)
-6 : 2102272 : 2^10,2053 (2102266)
-6 : 815634435 : 3,5,7,11,547,1291 (815634429)
6 : 8925 : 3,5^2,7,17 (8931)
6 : 32445 : 3^2,5,7,103 (32451)
6 : 442365 : 3,5,7,11,383 (442371)
-7 : 50 : 2,5^2 (43)
7 : 196 : 2^2,7^2 (203)
-8 : 22 : 2,11 (14)
-8 : 130 : 2,5,13 (122)
-8 : 184 : 2^3,23 (176)
-8 : 1012 : 2^2,11,23 (1004)
-8 : 2272 : 2^5,71 (2264)
-8 : 18904 : 2^3,17,139 (18896)
-8 : 33664 : 2^7,263 (33656)
-8 : 70564 : 2^2,13,23,59 (70556)
-8 : 85936 : 2^4,41,131 (85928)
-8 : 100804 : 2^2,11,29,79 (100796)
-8 : 391612 : 2^2,13,17,443 (391604)
-8 : 527872 : 2^9,1031 (527864)
-8 : 1090912 : 2^5,73,467 (1090904)
-8 : 17619844 : 2^2,11,37,79,137 (17619836)
8 : 56 : 2^3,7 (64)
8 : 368 : 2^4,23 (376)
8 : 836 : 2^2,11,19 (844)
8 : 11096 : 2^3,19,73 (11104)
8 : 17816 : 2^3,17,131 (17824)
8 : 45356 : 2^2,17,23,29 (45364)
8 : 77744 : 2^4,43,113 (77752)
8 : 91388 : 2^2,11,31,67 (91396)
8 : 128768 : 2^8,503 (128776)
8 : 254012 : 2^2,11,23,251 (254020)
8 : 388076 : 2^2,13,17,439 (388084)
8 : 2087936 : 2^10,2039 (2087944)
8 : 2291936 : 2^5,67,1069 (2291944)
8 : 13174976 : 2^6,139,1481 (13174984)
8 : 29465852 : 2^2,13,23,71,347 (29465860)
8 : 35021696 : 2^7,419,653 (35021704)
8 : 45335936 : 2^7,337,1051 (45335944)
8 : 120888092 : 2^2,13,23,61,1657 (120888100)
8 : 260378492 : 2^2,11,23,457,563 (260378500)
8 : 381236216 : 2^3,19^2,101,1307 (381236224)
8 : 775397948 : 2^2,13,17,661,1327 (775397956)
-10 : 21 : 3,7 (11)
-10 : 26 : 2,13 (16)
-10 : 68 : 2^2,17 (58)
-10 : 656 : 2^4,41 (646)
-10 : 2336 : 2^5,73 (2326)
-10 : 8768 : 2^6,137 (8758)
-10 : 133376 : 2^8,521 (133366)
-10 : 528896 : 2^9,1033 (528886)
10 : 40 : 2^3,5 (50)
10 : 1696 : 2^5,53 (1706)
10 : 518656 : 2^9,1013 (518666)
-11 : 244036 : 2^2,13^2,19^2 (244025)
-12 : 45 : 3^2,5 (33)
-12 : 76 : 2^2,19 (64)
-12 : 688 : 2^4,43 (676)
-12 : 8896 : 2^6,139 (8884)
-12 : 133888 : 2^8,523 (133876)
-12 : 537051136 : 2^14,32779 (537051124)
-14 : 27 : 3^3 (13)
-14 : 34 : 2,17 (20)
-14 : 232 : 2^3,29 (218)
-14 : 34432 : 2^7,269 (34418)
14 : 272 : 2^4,17 (286)
14 : 7232 : 2^6,113 (7246)
14 : 30848 : 2^7,241 (30862)
14 : 516608 : 2^9,1009 (516622)
14 : 134094848 : 2^13,16369 (134094862)
-16 : 38 : 2,19 (22)
-16 : 92 : 2^2,23 (76)
-16 : 170 : 2,5,17 (154)
-16 : 248 : 2^3,31 (232)
-16 : 752 : 2^4,47 (736)
-16 : 988 : 2^2,13,19 (972)
-16 : 2528 : 2^5,79 (2512)
-16 : 8648 : 2^3,23,47 (8632)
-16 : 12008 : 2^3,19,79 (11992)
-16 : 34688 : 2^7,271 (34672)
-16 : 63248 : 2^4,59,67 (63232)
-16 : 117808 : 2^4,37,199 (117792)
-16 : 526688 : 2^5,109,151 (526672)
-16 : 531968 : 2^9,1039 (531952)
-16 : 820808 : 2^3,37,47,59 (820792)
-16 : 1292768 : 2^5,71,569 (1292752)
-16 : 1495688 : 2^3,31,37,163 (1495672)
-16 : 2095208 : 2^3,23,59,193 (2095192)
-16 : 2112512 : 2^10,2063 (2112496)
-16 : 3477608 : 2^3,19,137,167 (3477592)
-16 : 4495808 : 2^6,199,353 (4495792)
-16 : 8419328 : 2^11,4111 (8419312)
-16 : 12026888 : 2^3,17,191,463 (12026872)
-16 : 13192768 : 2^6,139,1483 (13192752)
-16 : 16102808 : 2^3,17,167,709 (16102792)
-16 : 26347688 : 2^3,17,151,1283 (26347672)
-16 : 29322008 : 2^3,17,149,1447 (29321992)
-16 : 33653888 : 2^7,467,563 (33653872)
-16 : 169371008 : 2^7,269,4919 (169370992)
-16 : 173631608 : 2^3,17,137,9319 (173631592)
-16 : 293947648 : 2^8,787,1459 (293947632)
-16 : 537116672 : 2^14,32783 (537116656)
-16 : 883927808 : 2^8,557,6199 (883927792)
16 : 550 : 2,5^2,11 (566)
16 : 748 : 2^2,11,17 (764)
16 : 1504 : 2^5,47 (1520)
16 : 7192 : 2^3,29,31 (7208)
16 : 7912 : 2^3,23,43 (7928)
16 : 10792 : 2^3,19,71 (10808)
16 : 17272 : 2^3,17,127 (17288)
16 : 30592 : 2^7,239 (30608)
16 : 1713592 : 2^3,23,67,139 (1713608)
16 : 4526272 : 2^6,197,359 (4526288)
16 : 8353792 : 2^11,4079 (8353808)
16 : 9928792 : 2^3,19,83,787 (9928808)
16 : 11547352 : 2^3,17,197,431 (11547368)
16 : 17999992 : 2^3,19,79,1499 (18000008)
16 : 89283592 : 2^3,17,139,4723 (89283608)
16 : 173482552 : 2^3,17,137,9311 (173482568)
16 : 361702144 : 2^8,677,2087 (361702160)
16 : 1081850752 : 2^7,257,32887 (1081850768)
17 : 100 : 2^2,5^2 (117)
-18 : 33 : 3,11 (15)
-18 : 105 : 3,5,7 (87)
-18 : 33705 : 3^2,5,7,107 (33687)
-18 : 33624064 : 2^12,8209 (33624046)
18 : 208 : 2^4,13 (226)
18 : 6976 : 2^6,109 (6994)
18 : 8415 : 3^2,5,11,17 (8433)
18 : 31815 : 3^2,5,7,101 (31833)
18 : 351351 : 3^3,7,11,13^2 (351369)
18 : 2077696 : 2^10,2029 (2077714)
18 : 20487159 : 3^2,7,11,17,37,47 (20487177)
18 : 159030135 : 3^5,5,11,73,163 (159030153)
18 : 536559616 : 2^14,32749 (536559634)
-19 : 25 : 5^2 (6)
-19 : 2312 : 2^3,17^2 (2293)
19 : 36 : 2^2,3^2 (55)
-20 : 46 : 2,23 (26)
-20 : 154 : 2,7,11 (134)
-20 : 190 : 2,5,19 (170)
-20 : 2656 : 2^5,83 (2636)
-20 : 6490 : 2,5,11,59 (6470)
-20 : 44650 : 2,5^2,19,47 (44630)
-20 : 318250 : 2,5^3,19,67 (318230)
-20 : 1360810 : 2,5,11,89,139 (1360790)
-20 : 1503370 : 2,5,11,79,173 (1503350)
-20 : 1788490 : 2,5,11,71,229 (1788470)
-20 : 3214090 : 2,5,11,61,479 (3214070)
-20 : 103712410 : 2,5,17,29,109,193 (103712390)
20 : 176 : 2^4,11 (196)
20 : 1376 : 2^5,43 (1396)
20 : 3230 : 2,5,17,19 (3250)
20 : 3770 : 2,5,13,29 (3790)
20 : 6848 : 2^6,107 (6868)
20 : 114256 : 2^4,37,193 (114276)
20 : 125696 : 2^8,491 (125716)
20 : 544310 : 2,5,13,53,79 (544330)
20 : 561824 : 2^5,97,181 (561844)
20 : 740870 : 2,5,13,41,139 (740890)
20 : 2075648 : 2^10,2027 (2075668)
20 : 4199030 : 2,5,11,59,647 (4199050)
20 : 4607296 : 2^6,193,373 (4607316)
20 : 8436950 : 2,5^2,19,83,107 (8436970)
20 : 33468416 : 2^12,8171 (33468436)
20 : 134045696 : 2^13,16363 (134045716)
20 : 199272950 : 2,5^2,19,47,4463 (199272970)
20 : 624032630 : 2,5,13,47,109,937 (624032650)
20 : 1113445430 : 2,5,17,29,71,3181 (1113445450)
-22 : 35 : 5,7 (13)
-22 : 39 : 3,13 (17)
-22 : 63 : 3^2,7 (41)
-22 : 116 : 2^2,29 (94)
-22 : 296 : 2^3,37 (274)
-22 : 848 : 2^4,53 (826)
-22 : 9536 : 2^6,149 (9514)
-22 : 35456 : 2^7,277 (35434)
-22 : 2118656 : 2^10,2069 (2118634)
-22 : 537214976 : 2^14,32789 (537214954)
22 : 1312 : 2^5,41 (1334)
22 : 29824 : 2^7,233 (29846)
22 : 8341504 : 2^11,4073 (8341526)
22 : 134029312 : 2^13,16361 (134029334)
-24 : 124 : 2^2,31 (100)
-24 : 9664 : 2^6,151 (9640)
24 : 112 : 2^4,7 (136)
24 : 6592 : 2^6,103 (6616)
24 : 124672 : 2^8,487 (124696)
24 : 33452032 : 2^12,8167 (33452056)
-25 : 98 : 2,7^2 (73)
-26 : 58 : 2,29 (32)
-26 : 75 : 3,5^2 (49)
-26 : 328 : 2^3,41 (302)
-26 : 850 : 2,5^2,17 (824)
-26 : 1210 : 2,5,11^2 (1184)
-26 : 2848 : 2^5,89 (2822)
-26 : 35968 : 2^7,281 (35942)
-26 : 537088 : 2^9,1049 (537062)
26 : 80 : 2^4,5 (106)
26 : 1184 : 2^5,37 (1210)
26 : 6464 : 2^6,101 (6490)
26 : 29312 : 2^7,229 (29338)
26 : 78975 : 3^5,5^2,13 (79001)
26 : 510464 : 2^9,997 (510490)
26 : 557192 : 2^3,17^2,241 (557218)
-28 : 62 : 2,31 (34)
-28 : 182 : 2,7,13 (154)
-28 : 230 : 2,5,23 (202)
-28 : 344 : 2^3,43 (316)
-28 : 944 : 2^4,59 (916)
-28 : 6710 : 2,5,11,61 (6682)
-28 : 20264 : 2^3,17,149 (20236)
-28 : 36224 : 2^7,283 (36196)
-28 : 538112 : 2^9,1051 (538084)
-28 : 2085710 : 2,5,11,67,283 (2085682)
-28 : 14503550 : 2,5^2,17,113,151 (14503522)
-28 : 33665024 : 2^12,8219 (33664996)
-28 : 55328384 : 2^7,313,1381 (55328356)
-28 : 134438912 : 2^13,16411 (134438884)
-28 : 615206030 : 2,5,17,37,47,2081 (615206002)
-28 : 1082574464 : 2^7,257,32909 (1082574436)
28 : 48 : 2^4,3 (76)
28 : 2002 : 2,7,11,13 (2030)
28 : 5170 : 2,5,11,47 (5198)
28 : 29056 : 2^7,227 (29084)
28 : 133042 : 2,7,13,17,43 (133070)
28 : 114203776 : 2^7,277,3221 (114203804)
-30 : 51 : 3,17 (21)
-30 : 135 : 3^3,5 (105)
-30 : 148 : 2^2,37 (118)
-30 : 976 : 2^4,61 (946)
-30 : 10048 : 2^6,157 (10018)
-30 : 34335 : 3^2,5,7,109 (34305)
-30 : 138496 : 2^8,541 (138466)
-30 : 449295 : 3,5,7,11,389 (449265)
-30 : 33673216 : 2^12,8221 (33673186)
-30 : 537346048 : 2^14,32797 (537346018)
30 : 945 : 3^3,5,7 (975)
30 : 6208 : 2^6,97 (6238)
30 : 7425 : 3^3,5^2,11 (7455)
30 : 15028 : 2^2,13,17^2 (15058)
30 : 437745 : 3,5,7,11,379 (437775)
30 : 2065408 : 2^10,2017 (2065438)
30 : 3428368 : 2^4,47^2,97 (3428398)
30 : 20355825 : 3,5^2,7^2,29,191 (20355855)
30 : 33427456 : 2^12,8161 (33427486)
30 : 78524145 : 3^2,5,7,109,2287 (78524175)
31 : 15376 : 2^4,31^2 (15407)
-32 : 250 : 2,5^3 (218)
-32 : 376 : 2^3,47 (344)
-32 : 1276 : 2^2,11,29 (1244)
-32 : 12616 : 2^3,19,83 (12584)
-32 : 20536 : 2^3,17,151 (20504)
-32 : 396916 : 2^2,13,17,449 (396884)
-32 : 801376 : 2^5,79,317 (801344)
-32 : 1297312 : 2^5,71,571 (1297280)
-32 : 8452096 : 2^11,4127 (8452064)
-32 : 33721216 : 2^7,463,569 (33721184)
-32 : 40575616 : 2^7,359,883 (40575584)
-32 : 59376256 : 2^7,307,1511 (59376224)
-32 : 89397016 : 2^3,17,139,4729 (89396984)
-32 : 99523456 : 2^7,281,2767 (99523424)
-32 : 101556016 : 2^4,41,149,1039 (101555984)
-32 : 150441856 : 2^7,271,4337 (150441824)
-32 : 173706136 : 2^3,17,137,9323 (173706104)
-32 : 269096704 : 2^8,953,1103 (269096672)
-32 : 283417216 : 2^7,263,8419 (283417184)
-32 : 500101936 : 2^4,59,67,7907 (500101904)
-32 : 1082640256 : 2^7,257,32911 (1082640224)
32 : 572 : 2^2,11,13 (604)
32 : 992 : 2^5,31 (1024)
32 : 7544 : 2^3,23,41 (7576)
32 : 10184 : 2^3,19,67 (10216)
32 : 28544 : 2^7,223 (28576)
32 : 83312 : 2^4,41,127 (83344)
32 : 113072 : 2^4,37,191 (113104)
32 : 122624 : 2^8,479 (122656)
32 : 382772 : 2^2,13,17,433 (382804)
32 : 507392 : 2^9,991 (507424)
32 : 537248 : 2^5,103,163 (537280)
32 : 698528 : 2^5,83,263 (698560)
32 : 791264 : 2^5,79,313 (791296)
32 : 1081568 : 2^5,73,463 (1081600)
32 : 1279136 : 2^5,71,563 (1279168)
32 : 2154584 : 2^3,29,37,251 (2154616)
32 : 2279072 : 2^5,67,1063 (2279104)
32 : 5029184 : 2^6,179,439 (5029216)
32 : 15126992 : 2^4,67,103,137 (15127024)
32 : 29581424 : 2^4,47,139,283 (29581456)
32 : 74899952 : 2^4,47,103,967 (74899984)
32 : 89245784 : 2^3,17,139,4721 (89245816)
32 : 95327216 : 2^4,43,127,1091 (95327248)
32 : 307801856 : 2^8,751,1601 (307801888)
32 : 623799776 : 2^5,113,167,1033 (623799808)
32 : 712023296 : 2^8,571,4871 (712023328)
32 : 903230984 : 2^3,29,41,269,353 (903231016)
-34 : 57 : 3,19 (23)
-34 : 74 : 2,37 (40)
-34 : 164 : 2^2,41 (130)
-34 : 3104 : 2^5,97 (3070)
-34 : 2130944 : 2^10,2081 (2130910)
-34 : 8456192 : 2^11,4129 (8456158)
-34 : 134488064 : 2^13,16417 (134488030)
-34 : 537411584 : 2^14,32801 (537411550)
34 : 928 : 2^5,29 (962)
34 : 133931008 : 2^13,16349 (133931042)
-36 : 172 : 2^2,43 (136)
-36 : 1072 : 2^4,67 (1036)
-36 : 10432 : 2^6,163 (10396)
-36 : 140032 : 2^8,547 (139996)
-36 : 2132992 : 2^10,2083 (2132956)
-36 : 537444352 : 2^14,32803 (537444316)
36 : 2205 : 3^2,5,7^2 (2241)
36 : 2059264 : 2^10,2011 (2059300)
-37 : 484 : 2^2,11^2 (447)
-38 : 55 : 5,11 (17)
-38 : 82 : 2,41 (44)
-38 : 424 : 2^3,53 (386)
-38 : 3232 : 2^5,101 (3194)
-38 : 37504 : 2^7,293 (37466)
-38 : 543232 : 2^9,1061 (543194)
-38 : 8464384 : 2^11,4133 (8464346)
-38 : 134520832 : 2^13,16421 (134520794)
38 : 5696 : 2^6,89 (5734)
38 : 8308736 : 2^11,4057 (8308774)
39 : 162 : 2,3^4 (201)
-40 : 86 : 2,43 (46)
-40 : 188 : 2^2,47 (148)
-40 : 290 : 2,5,29 (250)
-40 : 950 : 2,5^2,19 (910)
-40 : 1136 : 2^4,71 (1096)
-40 : 1196 : 2^2,13,23 (1156)
-40 : 1364 : 2^2,11,31 (1324)
-40 : 3296 : 2^5,103 (3256)
-40 : 10688 : 2^6,167 (10648)
-40 : 260084 : 2^2,11,23,257 (260044)
-40 : 544256 : 2^9,1063 (544216)
-40 : 2137088 : 2^10,2087 (2137048)
-40 : 33714176 : 2^12,8231 (33714136)
-40 : 35221184 : 2^6,131,4201 (35221144)
-40 : 856444448 : 2^5,101,193,1373 (856444408)
40 : 736 : 2^5,23 (776)
40 : 381004 : 2^2,13,17,431 (381044)
40 : 503296 : 2^9,983 (503336)
40 : 2274784 : 2^5,67,1061 (2274824)
-41 : 49 : 7^2 (8)
-41 : 81 : 3^4 (40)
41 : 1352 : 2^3,13^2 (1393)
-42 : 69 : 3,23 (27)
-42 : 99 : 3^2,11 (57)
-42 : 165 : 3,5,11 (123)
-42 : 1168 : 2^4,73 (1126)
-42 : 1365 : 3,5,7,13 (1323)
-42 : 2139136 : 2^10,2089 (2139094)
-42 : 32062485 : 3,5,7,13,83,283 (32062443)
-42 : 33722368 : 2^12,8233 (33722326)
-42 : 132701205 : 3,5,7,13,67,1451 (132701163)
42 : 9555 : 3,5,7^2,13 (9597)
42 : 30555 : 3^2,5,7,97 (30597)
-44 : 94 : 2,47 (50)
-44 : 238 : 2,7,17 (194)
-44 : 310 : 2,5,31 (266)
-44 : 472 : 2^3,59 (428)
-44 : 3424 : 2^5,107 (3380)
-44 : 3910 : 2,5,17,23 (3866)
-44 : 4810 : 2,5,13,37 (4766)
-44 : 21352 : 2^3,17,157 (21308)
-44 : 2765092 : 2^2,11^2,29,197 (2765048)
-44 : 8476672 : 2^11,4139 (8476628)
-44 : 42305350 : 2,5^2,17,71,701 (42305306)
-44 : 134569984 : 2^13,16427 (134569940)
-44 : 1082837632 : 2^7,257,32917 (1082837588)
44 : 350 : 2,5^2,7 (394)
44 : 608 : 2^5,19 (652)
44 : 4730 : 2,5,11,43 (4774)
44 : 5312 : 2^6,83 (5356)
44 : 15368 : 2^3,17,113 (15412)
44 : 27008 : 2^7,211 (27052)
44 : 119552 : 2^8,467 (119596)
44 : 2051072 : 2^10,2003 (2051116)
44 : 8296448 : 2^11,4051 (8296492)
44 : 33370112 : 2^12,8147 (33370156)
44 : 133849088 : 2^13,16339 (133849132)
-46 : 65 : 5,13 (19)
-46 : 212 : 2^2,53 (166)
-46 : 488 : 2^3,61 (442)
-46 : 3488 : 2^5,109 (3442)
-46 : 11072 : 2^6,173 (11026)
-46 : 142592 : 2^8,557 (142546)
-46 : 547328 : 2^9,1069 (547282)
-46 : 33738752 : 2^12,8237 (33738706)
46 : 490 : 2,5,7^2 (536)
46 : 544 : 2^5,17 (590)
46 : 500224 : 2^9,977 (500270)
46 : 8292352 : 2^11,4049 (8292398)
-47 : 225 : 3^2,5^2 (178)
-48 : 1264 : 2^4,79 (1216)
48 : 60 : 2^2,3,5 (108)
48 : 5056 : 2^6,79 (5104)
48 : 118528 : 2^8,463 (118576)
48 : 2046976 : 2^10,1999 (2047024)
48 : 536068096 : 2^14,32719 (536068144)
0 : 6 : 2,3 (6)
0 : 28 : 2^2,7 (28)
0 : 496 : 2^4,31 (496)
0 : 8128 : 2^6,127 (8128)
0 : 33550336 : 2^12,8191 (33550336)
-1 : 4 : 2^2 (3)
-1 : 8 : 2^3 (7)
-1 : 16 : 2^4 (15)
-1 : 32 : 2^5 (31)
-1 : 64 : 2^6 (63)
-1 : 128 : 2^7 (127)
-1 : 256 : 2^8 (255)
-1 : 512 : 2^9 (511)
-1 : 1024 : 2^10 (1023)
-1 : 2048 : 2^11 (2047)
-1 : 4096 : 2^12 (4095)
-1 : 8192 : 2^13 (8191)
-1 : 16384 : 2^14 (16383)
-1 : 32768 : 2^15 (32767)
-1 : 65536 : 2^16 (65535)
-1 : 131072 : 2^17 (131071)
-1 : 262144 : 2^18 (262143)
-1 : 524288 : 2^19 (524287)
-1 : 1048576 : 2^20 (1048575)
-1 : 2097152 : 2^21 (2097151)
-1 : 4194304 : 2^22 (4194303)
-1 : 8388608 : 2^23 (8388607)
-1 : 16777216 : 2^24 (16777215)
-1 : 33554432 : 2^25 (33554431)
-1 : 67108864 : 2^26 (67108863)
-1 : 134217728 : 2^27 (134217727)
-1 : 268435456 : 2^28 (268435455)
-1 : 536870912 : 2^29 (536870911)
-1 : 1073741824 : 2^30 (1073741823)

As before, I've removed the composite factors to make it simpler to read, and attached the full list as a .csv file. The total of all factors is in parentheses.

 

While looking for numbers that are close to perfect, I've also been keeping note of the ones that are a long way away from it. The most "imperfect" number I've found so far is 1,102,701,600, which factorises to 2^5,3^4,5^2,7,11,13,17, for a total of 1,439 distinct factors summing to 4,614,182,496 - abundant by 3,511,480,896 ... I did think of posting the full list of factors here, but decided against it! ;)

[Turtle]

Normal programming resumed following a much-needed break :)

 

We've now trudged our way up to a billion. There are 477 numbers abundant/deficient by less than 50:

 

work above & beyond the call!! my genuine genuflection to your direction.

 

here's a breakdown of your reported counts-per-interval as ratios:

count/interval = ratio

372/10000000 = 0.0000372

443/250000000 = 0.000001772

452/500000000 = 0.0000000904

477/1000000000 = 0.000000477

 

getting leaner & leaner as a group. i think we should look at some of the individuals in the same ratio light, such as D16 & A16. i suspect their popularity is related & similar and/or scaled to the powers of 2. just a bump on that note again ; can we prove that all powers of 2 must be deficient by 1? could be useful. as i said earlier, i have only seen it asserted as fact, but no proofs. :shrug:

 

 

While looking for numbers that are close to perfect, I've also been keeping note of the ones that are a long way away from it. The most "imperfect" number I've found so far is 1,102,701,600, which factorises to 2^5,3^4,5^2,7,11,13,17, for a total of 1,439 distinct factors summing to 4,614,182,496 - abundant by 3,511,480,896 ... I did think of posting the full list of factors here, but decided against it! :hihi:

 

that would be one of my so called "phat numbers"; the ratio of 4614182496 to 1102701600 is 4.1844343891402714932126696832579. phat indeed by the standards of my intervals searched for phat numbers, but we did find that this ratio when taken, over the infinite set of natural numbers, has no limit. :eek: to infinity, and beyond!!! :doh: well, at a turtle's pace of course. . . . . . ;)

[Don Blazys]

Quoting Turtle:

Can we prove that all powers of 2 must be deficient by 1?

 

It's a "special case" of a more general formula.

 

If [math]p[/math] is prime, then the sum of the factors of [math]p^n[/math] is:

 

[math]\frac{p^n-1}{p-1}[/math]

 

Don.

[Turtle]

Quoting Turtle:

 

It's a "special case" of a more general formula.

 

If [math]p[/math] is prime, then the sum of the factors of [math]p^n[/math] is:

 

[math]\frac{p^n-1}{p-1}[/math]

 

Don.

 

just going to scratch paper the substitution for my eye

[math]\frac{2^n-1}{2-1}[/math]

 

for n=1; Σ = 1: x=2; Σ - x = -1

for n=2; Σ = 3: x=4; Σ - x = -1

for n=3; Σ = 7: x=8; Σ - x = -1

for n=4; Σ = 15: x=16; Σ - x = -1

.

.

.

 

how's that look? i was thinking to substitute this for the 2^n part of donk's found A16 & D16 & then try & concoct some rule for the rest of the factors. although i've seen it written that multiples of perfects and abundants are abundant, i have seen no such statement/rule invoking multiples of deficients. well... i can think it over while i hose out the trash cans. gotta run . . . . .

[Turtle]

... well... i can think it over while i hose out the trash cans. gotta run . . . . .

Damn the cans were filthy. Took 5 years to hose those puppies. Anyway, I saw Donk logged in a couple days back and since Craig took mention of another of my mathics I thought I'd see where we left off here. I reread the entire thread and re-downloaded a couple of Modest's files and got a somewhat hazy reconnection of thoughts thunk and thinkin' un-thunk. Seems we posed more questions than answered them and so in a conclusive sense it's not a dead thread.

 

The upshot is I have trepidations about stretching my brain as hard as I would need to pick this up again and so I thought I'd try and invoke similar trepidations & anxieties in y'all. Misery loves company. :edizzy:

 

PS You have by now noticed how long it takes this page to load and it's on account of the lengthy lists in Donk's posts. If we do pick up here we might get around that by putting the lists in spoilers. Also some Latex notation is returning errors and we might try and fix that too.

[Turtle]

...

can we prove that all powers of 2 must be deficient by 1? could be useful. as i said earlier, i have only seen it asserted as fact, but no proofs. :shrug:...

We can put this question to rest.

Types of Numbers

ALMOST PERFECT NUMBER

 

An almost perfect number is typically applied to the powers of 2 since the sum of the aliquot parts is 2ⁿ-1, or just 1 short of being a perfect number. It follows that any power of 2 is a deficient number.

When they say 'typically applied' they seem to imply that a power of 2 is sufficient for a deficiency of 1, but not necessary. Donk's list above to 500 million has no numbers deficient by 1 other than powers of 2. His list to 1 billion has no numbers deficient by 1 at all but it's not clear to me if he cut them out by specifying powers of 2 or specifying deficient by 1.

 

Oh lordy! What am I doing here!?

[CraigD]

We can put this question to rest.

...

can we prove that all powers of 2 must be deficient by 1? could be useful. as i said earlier, i have only seen it asserted as fact, but no proofs. :shrug:...

Types of Numbers

ALMOST PERFECT NUMBER

 

An almost perfect number is typically applied to the powers of 2 since the sum of the aliquot parts is 2ⁿ-1, or just 1 short of being a perfect number. It follows that any power of 2 is a deficient number.

 

Tappe (the fellow who wrote the mathgoodies.com article) doesn’t bother to prove the conjecture

2ⁿ-s(2ⁿ)=1

for all positive intergers n, I guess because he assumes it’s obviously true.

 

While not a neat, formal proof, here’s why it’s obvious to me:

Every power of two 2ⁿcan be written as a binary numeral beginning “1” and followed by n “0”s, and that that number minus 1 is written as a binary numeral consisting of n “1”s.

The proper divisors or 2ⁿ are 2⁰, … , 2^n-1.

[math]\sum_0^{n-1}[/math] is the same as the value of the binary numeral consisting of n

“1”s.

For example, 2⁸ = 256₁₀ = 100000000₂,

2⁸-1 = 11111111₂.

 

I was about to blurt out that only powers of 2 have interesting abundances, but taking a quick peek at the powers of 3 and their abundances:

3-1=2

9-4=5

27-13=14

81-40=41

...

129140163-64570081=64570082

suggests that

[math]\frac{3^n+1}{3^n-s(3^n)}=2[/math]

is true for all positive integers n

 

Nothing jumps out at me after a quick peek at the powers or 5, but I’m curious if there’s something remarkable about those I’m just not intuiting at a glance, and about the abundances of the powers of all the primes.

[Turtle]

We can put this question to rest.

Types of Numbers

 

When they say 'typically applied' they seem to imply that a power of 2 is sufficient for a deficiency of 1, but not necessary. Donk's list above to 500 million has no numbers deficient by 1 other than powers of 2. His list to 1 billion has no numbers deficient by 1 at all but it's not clear to me if he cut them out by specifying powers of 2 or specifying deficient by 1.

 

Oh lordy! What am I doing here!?

So after re-reading the thread a couple times, I see that I reported in post #15 that it is not known whether or not -other than 2ⁿs- any numbers deficient by 1 exist and that such numbers are termed 'least deficient'.

note: id:A000079 - OEIS Search Results

 

Least deficient or near-perfect numbers (i.e. n such that sigma(n)=A000203(n)=2n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 03 2004. Comment from Max Alekseyev (maxale(AT)gmail.com), Jan 26 2005: All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number not a power of 2. ...

 

I also see that we have uncovered no numbers other than 18 that are Abundant by 3. Mayhaps there is a term for them as well. :shrug:

 

Craig, I saw you eyeing our little party and wonder how did you miss it in the first place. :xparty: You know it tastes like caaaaannnndy. Any chance Craig that you can edit Donk's long lists into spoilers so it doesn't take so long to load this page? Much obliged.

 

And where's that Phillip & his pet python? Come out, come out wherever you are...