Deficient & Abundant Number Fun

[Turtle]

Brilliant.... as they say across the pond.

 

With some time in hand tomorrow, I'll see if I can get that list a bit longer. Hopefully I'll even be able to look at it :rolleyes: :doh: :hihi:

 

~modest

 

:hyper: i was able to open sorted_402192.txt in notepad if that mollifies you a bit. i'm @ 3,458,211 after searching overnight & no new A8's to be had. :cry:

[modest]

Ok. Hopefully we can confirm...

 

I've, unfortunately, only just now had a chance to kick-start this program. Watching how slow it's going though, I think we may need to compile it in C or maybe find a better factoring algorithm.

 

I'll let it run for the night.

 

~modest

[modest]

Argh!

 

Where's Donk when you need him... with his quick algorithms and fancy sorting :D

 

After running all night I got it up to a measly 794,820. Here are the -7 through 10's...

 

-7, 50, 1 2 5 10 25 
-6, 7, 1 
-6, 15, 1 3 5 
-6, 52, 1 2 4 13 26 
-6, 315, 1 3 5 7 9 15 21 35 45 63 105 
-6, 592, 1 2 4 8 16 37 74 148 296 
-6, 1155, 1 3 5 7 11 15 21 33 35 55 77 105 165 231 385 
-5, 9, 1 3 
-4, 5, 1 
-4, 14, 1 2 7 
-4, 44, 1 2 4 11 22 
-4, 110, 1 2 5 10 11 22 55 
-4, 152, 1 2 4 8 19 38 76 
-4, 884, 1 2 4 13 17 26 34 52 68 221 442 
-4, 2144, 1 2 4 8 16 32 67 134 268 536 1072 
-4, 8384, 1 2 4 8 16 32 64 131 262 524 1048 2096 4192 
-4, 18632, 1 2 4 8 17 34 68 136 137 274 548 1096 2329 4658 9316 
-4, 116624, 1 2 4 8 16 37 74 148 197 296 394 592 788 1576 3152 7289 14578 29156 58312 
-2, 3, 1 
-2, 10, 1 2 5 
-2, 136, 1 2 4 8 17 34 68 
-2, 32896, 1 2 4 8 16 32 64 128 257 514 1028 2056 4112 8224 16448 
-1, 2, 1 
-1, 4, 1 2 
-1, 8, 1 2 4 
-1, 16, 1 2 4 8 
-1, 32, 1 2 4 8 16 
-1, 64, 1 2 4 8 16 32 
-1, 128, 1 2 4 8 16 32 64 
-1, 256, 1 2 4 8 16 32 64 128 
-1, 512, 1 2 4 8 16 32 64 128 256 
-1, 1024, 1 2 4 8 16 32 64 128 256 512 
-1, 2048, 1 2 4 8 16 32 64 128 256 512 1024 
-1, 4096, 1 2 4 8 16 32 64 128 256 512 1024 2048 
-1, 8192, 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 
-1, 16384, 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 
-1, 32768, 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 
-1, 65536, 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 
-1, 131072, 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 
-1, 262144, 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 
-1, 524288, 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 
0, 1, 1 
0, 6, 1 2 3 
0, 28, 1 2 4 7 14 
0, 496, 1 2 4 8 16 31 62 124 248 
0, 8128, 1 2 4 8 16 32 64 127 254 508 1016 2032 4064 
2, 20, 1 2 4 5 10 
2, 104, 1 2 4 8 13 26 52 
2, 464, 1 2 4 8 16 29 58 116 232 
2, 650, 1 2 5 10 13 25 26 50 65 130 325 
2, 1952, 1 2 4 8 16 32 61 122 244 488 976 
2, 130304, 1 2 4 8 16 32 64 128 256 509 1018 2036 4072 8144 16288 32576 65152 
2, 522752, 1 2 4 8 16 32 64 128 256 512 1021 2042 4084 8168 16336 32672 65344 130688 261376 
3, 18, 1 2 3 6 9 
4, 12, 1 2 3 4 6 
4, 70, 1 2 5 7 10 14 35 
4, 88, 1 2 4 8 11 22 44 
4, 1888, 1 2 4 8 16 32 59 118 236 472 944 
4, 4030, 1 2 5 10 13 26 31 62 65 130 155 310 403 806 2015 
4, 5830, 1 2 5 10 11 22 53 55 106 110 265 530 583 1166 2915 
4, 32128, 1 2 4 8 16 32 64 128 251 502 1004 2008 4016 8032 16064 
4, 521728, 1 2 4 8 16 32 64 128 256 512 1019 2038 4076 8152 16304 32608 65216 130432 260864 
6, 8925, 1 3 5 7 15 17 21 25 35 51 75 85 105 119 175 255 357 425 525 595 1275 1785 2975 
6, 32445, 1 3 5 7 9 15 21 35 45 63 103 105 309 315 515 721 927 1545 2163 3605 4635 6489 10815 
6, 442365, 1 3 5 7 11 15 21 33 35 55 77 105 165 231 383 385 1149 1155 1915 2681 4213 5745 8043 12639 13405 21065 29491 40215 63195 88473 147455 
7, 196, 1 2 4 7 14 28 49 98 
8, 56, 1 2 4 7 8 14 28 
8, 368, 1 2 4 8 16 23 46 92 184 
8, 836, 1 2 4 11 19 22 38 44 76 209 418 
8, 11096, 1 2 4 8 19 38 73 76 146 152 292 584 1387 2774 5548 
8, 17816, 1 2 4 8 17 34 68 131 136 262 524 1048 2227 4454 8908 
8, 45356, 1 2 4 17 23 29 34 46 58 68 92 116 391 493 667 782 986 1334 1564 1972 2668 11339 22678 
8, 77744, 1 2 4 8 16 43 86 113 172 226 344 452 688 904 1808 4859 9718 19436 38872 
8, 91388, 1 2 4 11 22 31 44 62 67 124 134 268 341 682 737 1364 1474 2077 2948 4154 8308 22847 45694 
8, 128768, 1 2 4 8 16 32 64 128 256 503 1006 2012 4024 8048 16096 32192 64384 
8, 254012, 1 2 4 11 22 23 44 46 92 251 253 502 506 1004 1012 2761 5522 5773 11044 11546 23092 63503 127006 
8, 388076, 1 2 4 13 17 26 34 52 68 221 439 442 878 884 1756 5707 7463 11414 14926 22828 29852 97019 194038 
10, 40, 1 2 4 5 8 10 20 
10, 1696, 1 2 4 8 16 32 53 106 212 424 848 
10, 518656, 1 2 4 8 16 32 64 128 256 512 1013 2026 4052 8104 16208 32416 64832 129664 259328 

 

http://www.box.net/shared/t8xr5yrihj

 

It's 'bout 20 mb zipped. I tried to get it to a million, but it was getting exponentially slower. What program did you write yours in, T?

 

~modest

[Donk]

Argh!

 

Where's Donk when you need him... with his quick algorithms and fancy sorting :D

 

After running all night I got it up to a measly 794,820. Here are the -7 through 10's...

Sigh... I've been watching from the sidelines with great interest, telling myself "NO! Keep away! You have a busy, fulfilled life with no time for fun & games!!!"

 

Believe it or no, I decided this morning that I'd have to get involved. Mapped out a plan of attack while strolling to work, jotted down some notes when I got there... I should be able to sort it over the weekend. :D

 

I was thinking of a preliminary "counting" run, to see how the deficiencies & abundances fell, then the full programme would output separate files, one file for each of the common ones, and an outliers file for the values that don't crop up often enough to warrant a file to themselves. The plan is to get up to at least a billion, and probably 3 billion.

 

Format would be one line per number, in the form:

 

100 : 1 2 4 5 10 20 25 50 - 117 (+17)

 

The colon, dash and parens could easily be search-and-replaced into tabs ready for import into excel, making it easy to sort the outliers.

 

That's my thoughts. But you guys are the experts - what would you like?

[Turtle]

Argh!

 

Where's Donk when you need him... with his quick algorithms and fancy sorting :D

 

After running all night I got it up to a measly 794,820.

...

sorted_794820.zip - File Shared from Box.net - Free Online File Storage

 

It's 'bout 20 mb zipped. I tried to get it to a million, but it was getting exponentially slower. What program did you write yours in, T?

 

~modest

 

exponentially slower is a tortoise standard after all. :lol: i prolly wrote this before, but i'll tell what i'm using as there is nothing like laughter in the face of exponential slowatudeinality. i am using an old Borland DOS product from the late 80's early 90's called Turbo Basic. i originally ran it on a dual 5.25 floppy 8088 8mghz machine, but now have it running in a DOS window on an old Windows 95 machine kickin' out 200 mghz. no really.

 

the machine has no internet, no writable cd drive, & no USB's. :eek2: :doh: :hyper: i collect my data visually from the screen and write it in a notebook & then copy it over here. i'd say the reason i zipped ahead is that i was searching only for abundant-by-8's. :smart: i ran it to 5,000,000 today then stopped; no new A8's to report in that interval.

 

Sigh... I've been watching from the sidelines with great interest, telling myself "NO! Keep away! You have a busy, fulfilled life with no time for fun & games!!!"

 

Believe it or no, I decided this morning that I'd have to get involved. Mapped out a plan of attack while strolling to work, jotted down some notes when I got there... I should be able to sort it over the weekend.

 

I was thinking of a preliminary "counting" run, to see how the deficiencies & abundances fell, then the full programme would output separate files, one file for each of the common ones, and an outliers file for the values that don't crop up often enough to warrant a file to themselves. The plan is to get up to at least a billion, and probably 3 billion.

 

Format would be one line per number, in the form:

 

100 : 1 2 4 5 10 20 25 50 - 117 (+17)

 

The colon, dash and parens could easily be search-and-replaced into tabs ready for import into excel, making it easy to sort the outliers.

 

That's my thoughts. But you guys are the experts - what would you like?

 

:bow: it tastes like caaannnndyyy! :D muahahahaha :D

 

your plan is lookin' good. :D maybe have an option for including the prime factorization in the line? some parts are edible. ;)

 

let's make some charcoal!

[Turtle]

i resumed searching for A8's & i'm out to 6346000+ and no finds. i always get sketchy when i got nothin'; thinking i may have made an inadvertant typo in the code & not really searching at all. :doh: you guys gonna check me though so onward to infinity & beyond!

[modest]

Sigh... I've been watching from the sidelines with great interest, telling myself "NO! Keep away! You have a busy, fulfilled life with no time for fun & games!!!"

 

Believe it or no, I decided this morning that I'd have to get involved. Mapped out a plan of attack while strolling to work, jotted down some notes when I got there... I should be able to sort it over the weekend.

 

Alright

 

I was thinking of a preliminary "counting" run, to see how the deficiencies & abundances fell, then the full programme would output separate files, one file for each of the common ones, and an outliers file for the values that don't crop up often enough to warrant a file to themselves. The plan is to get up to at least a billion, and probably 3 billion.

 

That's what I'm talking about.

 

Can you remind me what programming language you use. I keep asking everybody that, but my C compiler crapped out on me a couple months ago and I'm fixing to throw Perl out the window

 

~modest

[Donk]

Can you remind me what programming language you use. I keep asking everybody that, but my C compiler crapped out on me a couple months ago and I'm fixing to throw Perl out the window

Currently I'm using Qbasic64. It's being developed by a group of Basic enthusiasts, and there are still a few bugs/issues with it, but it does pretty much what I need it to, and very much faster than the QB45 I was using last year.

 

I'm also looking at programming in pure assembler, something I haven't needed to do for over 30 years. Back then, the entire instruction set description fitted on to a single page. Now I'm ploughing through something like 10,000 pages of badly-written and self-contradictory description. If I can get it right, it'll be ideal for number theory work - many iterations of a very small code block should go much faster. But don't hold your breath

[Donk]

Everything seems to be working... the routine's been running about a half hour so far, and is up to 10,000,000+

 

I'm splitting it into 3 files:

Abundant-by 12

-20 to +20 (less the twelves)

The rest.

 

I had a quick look at abundant-by-2, and it's found

8382464 : 2^11,4093 : 1,2,4,8,16,32,64,128,256,512,1024,2048,4093,8186,16372,32744,65488,130976,261952,523904,1047808,2095616,4191232 : (8382466) : 2

 

The format is number, then prime factors, then all factors, then total of factors, then abundant/deficient.

 

Nothing new - it's one of the numbers on the OP - but at least it's tested the routine.

 

ps - just passing 13 million. Time for bed :)

[Donk]

I took a look at things when I finally got out of bed today, and decided to stop & restart. I have 150 gigabytes of spare disc space on this machine, and the routine looked like using all of it and then some!

 

Abundant/deficient by 20 or fewer wasn't the problem. Lots there, but manageable. It was the vast majority outside that range that caused the problem, so I decided to throw away any number abundant/deficient by more than 500.

 

Taking a look at the data collected so far, I've added abundant-by-56 to the file collecting abundant-by-12. As post#28 in this thread shows, numbers of the form 6*p are abundant by 12. This is clearly because 6 is a perfect number. The same logic applies to the other perfect numbers, so I'm segregating 56 (2x28) as well.

 

Part of the programme involves creating an array of all the factors, so I wanted an idea of how large that array would have to be. I set up a fourth output file containing all numbers with more than 300 factors. So far, the largest I've found has 719 factors (up to around 75 million). Only there's not just one of them. The list has three:

 

61261200:2^4, 3^2, 5^2, 7, 11, 13, 17:1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 
21, 22, 24, 25, 26, 28, 30, 33, 34, 35, 36, 39, 40, 42, 44, 45, 48, 50, 51, 52, 55, 56, 60, 63, 65, 66, 68, 
70, 72, 75, 77, 78, 80, 84, 85, 88, 90, 91, 99, 100, 102, 104, 105, 110, 112, 117, 119, 120, 126, 130, 
132, 136, 140, 143, 144, 150, 153, 154, 156, 165, 168, 170, 175, 176, 180, 182, 187, 195, 198, 200, 
204, 208, 210, 220, 221, 225, 231, 234, 238, 240, 252, 255, 260, 264, 272, 273, 275, 280, 286, 300, 
306, 308, 312, 315, 325, 330, 336, 340, 350, 357, 360, 364, 374, 385, 390, 396, 400, 408, 420, 425, 
429, 440, 442, 450, 455, 462, 468, 476, 495, 504, 510, 520, 525, 528, 546, 550, 560, 561, 572, 585, 
595, 600, 612, 616, 624, 630, 650, 660, 663, 680, 693, 700, 714, 715, 720, 728, 748, 765, 770, 780, 
792, 816, 819, 825, 840, 850, 858, 880, 884, 900, 910, 924, 935, 936, 952, 975, 990, 1001, 1008, 1020, 
1040, 1050, 1071, 1092, 1100, 1105, 1122, 1144, 1155, 1170, 1190, 1200, 1224, 1232, 1260, 1275, 1287, 
1300, 1309, 1320, 1326, 1360, 1365, 1386, 1400, 1428, 1430, 1456, 1496, 1530, 1540, 1547, 1560, 1575, 
1584, 1638, 1650, 1680, 1683, 1700, 1716, 1768, 1785, 1800, 1820, 1848, 1870, 1872, 1904, 1925, 
1950, 1980, 1989, 2002, 2040, 2100, 2142, 2145, 2184, 2200, 2210, 2244, 2275, 2288, 2310, 2340, 
2380, 2431, 2448, 2475, 2520, 2550, 2574, 2600, 2618, 2640, 2652, 2730, 2772, 2800, 2805, 2856, 
2860, 2925, 2975, 2992, 3003, 3060, 3080, 3094, 3120, 3150, 3276, 3300, 3315, 3366, 3400, 3432, 
3465, 3536, 3570, 3575, 3600, 3640, 3696, 3740, 3825, 3850, 3900, 3927, 3960, 3978, 4004, 4080, 
4095, 4200, 4284, 4290, 4368, 4400, 4420, 4488, 4550, 4620, 4641, 4675, 4680, 4760, 4862, 4950, 
5005, 5040, 5100, 5148, 5200, 5236, 5304, 5355, 5460, 5525, 5544, 5610, 5712, 5720, 5775, 5850, 
5950, 6006, 6120, 6160, 6188, 6300, 6435, 6545, 6552, 6600, 6630, 6732, 6800, 6825, 6864, 6930, 
7140, 7150, 7280, 7293, 7480, 7650, 7700, 7735, 7800, 7854, 7920, 7956, 8008, 8190, 8400, 8415, 
8568, 8580, 8840, 8925, 8976, 9009, 9100, 9240, 9282, 9350, 9360, 9520, 9724, 9900, 9945, 10010, 
10200, 10296, 10472, 10608, 10710, 10725, 10920, 11050, 11088, 11220, 11440, 11550, 11700, 
11781, 11900, 12012, 12155, 12240, 12376, 12600, 12870, 13090, 13104, 13200, 13260, 13464, 
13650, 13860, 13923, 14025, 14280, 14300, 14586, 14960, 15015, 15300, 15400, 15470, 15600, 
15708, 15912, 16016, 16380, 16575, 16830, 17017, 17136, 17160, 17325, 17680, 17850, 18018, 
18200, 18480, 18564, 18700, 19448, 19635, 19800, 19890, 20020, 20400, 20475, 20592, 20944, 
21420, 21450, 21840, 21879, 22100, 22440, 23100, 23205, 23400, 23562, 23800, 24024, 24310, 
24752, 25025, 25200, 25740, 26180, 26520, 26775, 26928, 27300, 27720, 27846, 28050, 28560, 
28600, 29172, 30030, 30600, 30800, 30940, 31416, 31824, 32175, 32725, 32760, 33150, 33660, 
34034, 34320, 34650, 35700, 36036, 36400, 36465, 37128, 37400, 38675, 38896, 39270, 39600, 
39780, 40040, 40950, 42075, 42840, 42900, 43758, 44200, 44880, 45045, 46200, 46410, 46800, 
47124, 47600, 48048, 48620, 49725, 50050, 51051, 51480, 52360, 53040, 53550, 54600, 55440, 
55692, 56100, 57200, 58344, 58905, 60060, 60775, 61200, 61880, 62832, 64350, 65450, 65520, 
66300, 67320, 68068, 69300, 69615, 71400, 72072, 72930, 74256, 74800, 75075, 77350, 78540, 
79560, 80080, 81900, 84150, 85085, 85680, 85800, 87516, 88400, 90090, 92400, 92820, 94248, 
97240, 98175, 99450, 100100, 102102, 102960, 104720, 107100, 109200, 109395, 111384, 112200, 
116025, 116688, 117810, 120120, 121550, 123760, 128700, 130900, 132600, 134640, 136136, 138600, 
139230, 142800, 144144, 145860, 150150, 153153, 154700, 157080, 159120, 163800, 168300, 170170, 
171600, 175032, 180180, 182325, 185640, 188496, 194480, 196350, 198900, 200200, 204204, 214200, 
218790, 222768, 224400, 225225, 232050, 235620, 240240, 243100, 255255, 257400, 261800, 265200, 
272272, 277200, 278460, 291720, 294525, 300300, 306306, 309400, 314160, 327600, 336600, 340340, 
348075, 350064, 360360, 364650, 371280, 392700, 397800, 400400, 408408, 425425, 428400, 437580, 
450450, 464100, 471240, 486200, 510510, 514800, 523600, 546975, 556920, 583440, 589050, 600600, 
612612, 618800, 673200, 680680, 696150, 720720, 729300, 765765, 785400, 795600, 816816, 850850, 
875160, 900900, 928200, 942480, 972400, 1021020, 1093950, 1113840, 1178100, 1201200, 1225224, 
1276275, 1361360, 1392300, 1458600, 1531530, 1570800, 1701700, 1750320, 1801800, 1856400, 2042040, 
2187900, 2356200, 2450448, 2552550, 2784600, 2917200, 3063060, 3403400, 3603600, 3828825, 4084080, 
4375800, 4712400, 5105100, 5569200, 6126120, 6806800, 7657650, 8751600, 10210200, 12252240, 
15315300, 20420400, 30630600:(240969456):179,708,256

64864800:2^5, 3^4, 5^2, 7, 11, 13:1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 
24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 
75, 77, 78, 80, 81, 84, 88, 90, 91, 96, 99, 100, 104, 105, 108, 110, 112, 117, 120, 126, 130, 132, 
135, 140, 143, 144, 150, 154, 156, 160, 162, 165, 168, 175, 176, 180, 182, 189, 195, 198, 200, 208, 
210, 216, 220, 224, 225, 231, 234, 240, 252, 260, 264, 270, 273, 275, 280, 286, 288, 297, 300, 308, 
312, 315, 324, 325, 330, 336, 350, 351, 352, 360, 364, 378, 385, 390, 396, 400, 405, 416, 420, 429, 
432, 440, 450, 455, 462, 468, 480, 495, 504, 520, 525, 528, 540, 546, 550, 560, 567, 572, 585, 594, 
600, 616, 624, 630, 648, 650, 660, 672, 675, 693, 700, 702, 715, 720, 728, 756, 770, 780, 792, 800, 
810, 819, 825, 840, 858, 864, 880, 891, 900, 910, 924, 936, 945, 975, 990, 1001, 1008, 1040, 1050, 
1053, 1056, 1080, 1092, 1100, 1120, 1134, 1144, 1155, 1170, 1188, 1200, 1232, 1248, 1260, 1287, 
1296, 1300, 1320, 1350, 1365, 1386, 1400, 1404, 1430, 1440, 1456, 1485, 1512, 1540, 1560, 1575, 
1584, 1620, 1638, 1650, 1680, 1716, 1755, 1760, 1782, 1800, 1820, 1848, 1872, 1890, 1925, 1950, 
1980, 2002, 2016, 2025, 2079, 2080, 2100, 2106, 2145, 2160, 2184, 2200, 2268, 2275, 2288, 2310, 
2340, 2376, 2400, 2457, 2464, 2475, 2520, 2574, 2592, 2600, 2640, 2700, 2730, 2772, 2800, 2808, 
2835, 2860, 2912, 2925, 2970, 3003, 3024, 3080, 3120, 3150, 3168, 3240, 3276, 3300, 3360, 3432, 
3465, 3510, 3564, 3575, 3600, 3640, 3696, 3744, 3780, 3850, 3861, 3900, 3960, 4004, 4050, 4095, 
4158, 4200, 4212, 4290, 4320, 4368, 4400, 4455, 4536, 4550, 4576, 4620, 4680, 4725, 4752, 4914, 
4950, 5005, 5040, 5148, 5200, 5265, 5280, 5400, 5460, 5544, 5600, 5616, 5670, 5720, 5775, 5850, 
5940, 6006, 6048, 6160, 6237, 6240, 6300, 6435, 6480, 6552, 6600, 6825, 6864, 6930, 7020, 7128, 
7150, 7200, 7280, 7371, 7392, 7425, 7560, 7700, 7722, 7800, 7920, 8008, 8100, 8190, 8316, 8400, 
8424, 8580, 8736, 8775, 8800, 8910, 9009, 9072, 9100, 9240, 9360, 9450, 9504, 9828, 9900, 10010, 
10080, 10296, 10395, 10400, 10530, 10725, 10800, 10920, 11088, 11232, 11340, 11440, 11550, 
11583, 11700, 11880, 12012, 12285, 12320, 12474, 12600, 12870, 12960, 13104, 13200, 13650, 
13728, 13860, 14040, 14175, 14256, 14300, 14560, 14742, 14850, 15015, 15120, 15400, 15444, 
15600, 15840, 16016, 16200, 16380, 16632, 16800, 16848, 17160, 17325, 17550, 17820, 18018, 
18144, 18200, 18480, 18720, 18900, 19305, 19656, 19800, 20020, 20475, 20592, 20790, 21060, 
21450, 21600, 21840, 22176, 22275, 22680, 22880, 23100, 23166, 23400, 23760, 24024, 24570, 
24948, 25025, 25200, 25740, 26208, 26325, 26400, 27027, 27300, 27720, 28080, 28350, 28512, 
28600, 29484, 29700, 30030, 30240, 30800, 30888, 31185, 31200, 32032, 32175, 32400, 32760, 
33264, 33696, 34320, 34650, 35100, 35640, 36036, 36400, 36855, 36960, 37800, 38610, 39312, 
39600, 40040, 40950, 41184, 41580, 42120, 42900, 43680, 44550, 45045, 45360, 46200, 46332, 
46800, 47520, 48048, 49140, 49896, 50050, 50400, 51480, 51975, 52650, 54054, 54600, 55440, 
56160, 56700, 57200, 57915, 58968, 59400, 60060, 61425, 61600, 61776, 62370, 64350, 64800, 
65520, 66528, 68640, 69300, 70200, 71280, 72072, 72800, 73710, 75075, 75600, 77220, 78624, 
79200, 80080, 81081, 81900, 83160, 84240, 85800, 89100, 90090, 90720, 92400, 92664, 93600, 
96096, 96525, 98280, 99792, 100100, 102960, 103950, 105300, 108108, 109200, 110880, 113400, 114400, 
115830, 117936, 118800, 120120, 122850, 123552, 124740, 128700, 131040, 135135, 138600, 140400, 
142560, 144144, 147420, 150150, 151200, 154440, 155925, 160160, 162162, 163800, 166320, 168480, 
171600, 178200, 180180, 184275, 184800, 185328, 193050, 196560, 199584, 200200, 205920, 207900, 
210600, 216216, 218400, 225225, 226800, 231660, 235872, 237600, 240240, 245700, 249480, 257400, 
270270, 277200, 280800, 288288, 289575, 294840, 300300, 308880, 311850, 324324, 327600, 332640, 
343200, 356400, 360360, 368550, 370656, 386100, 393120, 400400, 405405, 415800, 421200, 432432, 
450450, 453600, 463320, 480480, 491400, 498960, 514800, 540540, 554400, 579150, 589680, 600600, 
617760, 623700, 648648, 655200, 675675, 712800, 720720, 737100, 772200, 800800, 810810, 831600, 
842400, 864864, 900900, 926640, 982800, 997920, 1029600, 1081080, 1158300, 1179360, 1201200, 1247400, 
1297296, 1351350, 1441440, 1474200, 1544400, 1621620, 1663200, 1801800, 1853280, 1965600, 2027025, 
2162160, 2316600, 2402400, 2494800, 2594592, 2702700, 2948400, 3088800, 
3243240, 3603600, 4054050, 4324320, 4633200, 4989600, 5405400, 5896800, 6486480, 7207200, 8108100, 
9266400, 10810800, 12972960, 16216200, 21621600, 32432400:(252739872):187,875,072

68468400:2^4, 3^2, 5^2, 7, 11, 13, 19:1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 
21, 22, 24, 25, 26, 28, 30, 33, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 55, 56, 57, 60, 63, 65, 66, 70, 
72, 75, 76, 77, 78, 80, 84, 88, 90, 91, 95, 99, 100, 104, 105, 110, 112, 114, 117, 120, 126, 130, 132, 
133, 140, 143, 144, 150, 152, 154, 156, 165, 168, 171, 175, 176, 180, 182, 190, 195, 198, 200, 208, 
209, 210, 220, 225, 228, 231, 234, 240, 247, 252, 260, 264, 266, 273, 275, 280, 285, 286, 300, 304, 
308, 312, 315, 325, 330, 336, 342, 350, 360, 364, 380, 385, 390, 396, 399, 400, 418, 420, 429, 440, 
450, 455, 456, 462, 468, 475, 494, 495, 504, 520, 525, 528, 532, 546, 550, 560, 570, 572, 585, 600, 
616, 624, 627, 630, 650, 660, 665, 684, 693, 700, 715, 720, 728, 741, 760, 770, 780, 792, 798, 819, 
825, 836, 840, 855, 858, 880, 900, 910, 912, 924, 936, 950, 975, 988, 990, 1001, 1008, 1040, 1045, 
1050, 1064, 1092, 1100, 1140, 1144, 1155, 1170, 1197, 1200, 1232, 1235, 1254, 1260, 1287, 1300, 
1320, 1330, 1365, 1368, 1386, 1400, 1425, 1430, 1456, 1463, 1482, 1520, 1540, 1560, 1575, 1584, 
1596, 1638, 1650, 1672, 1680, 1710, 1716, 1729, 1800, 1820, 1848, 1872, 1881, 1900, 1925, 1950, 
1976, 1980, 1995, 2002, 2090, 2100, 2128, 2145, 2184, 2200, 2223, 2275, 2280, 2288, 2310, 2340, 
2394, 2470, 2475, 2508, 2520, 2574, 2600, 2640, 2660, 2717, 2730, 2736, 2772, 2800, 2850, 2860, 
2925, 2926, 2964, 3003, 3080, 3120, 3135, 3150, 3192, 3276, 3300, 3325, 3344, 3420, 3432, 3458, 
3465, 3575, 3600, 3640, 3696, 3705, 3762, 3800, 3850, 3900, 3952, 3960, 3990, 4004, 4095, 4180, 
4200, 4275, 4290, 4368, 4389, 4400, 4446, 4550, 4560, 4620, 4680, 4788, 4940, 4950, 5005, 5016, 
5040, 5148, 5187, 5200, 5225, 5320, 5434, 5460, 5544, 5700, 5720, 5775, 5850, 5852, 5928, 5985, 
6006, 6160, 6175, 6270, 6300, 6384, 6435, 6552, 6600, 6650, 6825, 6840, 6864, 6916, 6930, 7150, 
7280, 7315, 7410, 7524, 7600, 7700, 7800, 7920, 7980, 8008, 8151, 8190, 8360, 8400, 8550, 8580, 
8645, 8778, 8892, 9009, 9100, 9240, 9360, 9405, 9576, 9880, 9900, 9975, 10010, 10032, 10296, 
10374, 10450, 10640, 10725, 10868, 10920, 11088, 11115, 11400, 11440, 11550, 11700, 11704, 
11856, 11970, 12012, 12350, 12540, 12600, 12870, 13104, 13167, 13200, 13300, 13585, 13650, 
13680, 13832, 13860, 14300, 14630, 14820, 15015, 15048, 15400, 15561, 15600, 15675, 15960, 
16016, 16302, 16380, 16720, 17100, 17160, 17290, 17325, 17556, 17784, 18018, 18200, 18480, 
18525, 18810, 19019, 19152, 19760, 19800, 19950, 20020, 20475, 20592, 20748, 20900, 21450, 
21736, 21840, 21945, 22230, 22800, 23100, 23400, 23408, 23940, 24024, 24453, 24700, 25025, 
25080, 25200, 25740, 25935, 26334, 26600, 27170, 27300, 27664, 27720, 28600, 29260, 29640, 
29925, 30030, 30096, 30800, 31122, 31350, 31920, 32175, 32604, 32760, 34200, 34320, 34580, 
34650, 35112, 35568, 36036, 36400, 36575, 37050, 37620, 38038, 39600, 39900, 40040, 40755, 
40950, 41496, 41800, 42900, 43225, 43472, 43890, 44460, 45045, 46200, 46800, 47025, 47880, 
48048, 48906, 49400, 50050, 50160, 51480, 51870, 52668, 53200, 54340, 54600, 55440, 55575, 
57057, 57200, 58520, 59280, 59850, 60060, 62244, 62700, 64350, 65208, 65520, 65835, 67925, 
68400, 69160, 69300, 70224, 72072, 73150, 74100, 75075, 75240, 76076, 77805, 79800, 80080, 
81510, 81900, 82992, 83600, 85800, 86450, 87780, 88920, 90090, 92400, 94050, 95095, 95760, 
97812, 98800, 100100, 102960, 103740, 105336, 108680, 109200, 109725, 111150, 114114, 
117040, 119700, 120120, 122265, 124488, 125400, 128700, 129675, 130416, 131670, 135850, 
138320, 138600, 144144, 146300, 148200, 150150, 150480, 152152, 155610, 159600, 163020, 
163800, 171171, 171600, 172900, 175560, 177840, 180180, 188100, 190190, 195624, 200200, 
203775, 207480, 210672, 217360, 219450, 222300, 225225, 228228, 239400, 240240, 244530, 
248976, 250800, 257400, 259350, 263340, 271700, 277200, 285285, 292600, 296400, 300300, 
304304, 311220, 326040, 327600, 329175, 342342, 345800, 351120, 360360, 376200, 380380, 
389025, 391248, 400400, 407550, 414960, 438900, 444600, 450450, 456456, 475475, 478800, 
489060, 514800, 518700, 526680, 543400, 570570, 585200, 600600, 611325, 622440, 652080, 
658350, 684684, 691600, 720720, 752400, 760760, 778050, 815100, 855855, 877800, 889200, 
900900, 912912, 950950, 978120, 1037400, 1053360, 1086800, 1141140, 1201200, 1222650, 
1244880, 1316700, 1369368, 1426425, 1521520, 1556100, 1630200, 1711710, 1755600, 1801800, 
1901900, 1956240, 2074800, 2282280, 2445300, 2633400, 2738736, 2852850, 3112200, 3260400, 
3423420, 3603600, 3803800, 4279275, 4564560, 4890600, 5266800, 5705700, 6224400, 6846840, 
7607600, 8558550, 9781200, 11411400, 13693680, 17117100, 22822800, 
34234200:(267343440):198,875,040

Below those 3, the next highest number of factors is 671 - 10 of those. Here's the full list:

719	3
671	10
647	3
639	15
629	1
599	12
575	151
559	17
539	37
527	6
511	127
503	117
499	1
485	11
479	615
467	1
449	21
447	244
440	1
439	6
431	926
419	103
415	4
404	9
399	300
395	37
391	22
389	1
383	3277
377	70
374	1
359	1937
351	104
350	1
349	7
335	2055
329	17
323	757
319	3937
314	16
311	64
307	7

Can anyone figure out why the number of factors clusters like that? 3937 have exactly 319 factors, 3277 numbers have 383 ... very odd, but we like odd, don't we! :)

[modest]

Can anyone figure out why the number of factors clusters like that? 3937 have exactly 319 factors, 3277 numbers have 383 ... very odd, but we like odd, don't we! :)

 

I do not know why off hand, but I can, I think, confirm the pattern. The divisor function σ₀(n) counts the number of divisors of n. When the function is scatter-plotted, there are clear bands meaning that certain numbers of divisors are prevalent.

 

-

A000005

 

~modest

[Turtle]

I took a look at things when I finally got out of bed today, and decided to stop & restart. I have 150 gigabytes of spare disc space on this machine, and the routine looked like using all of it and then some!

 

Abundant/deficient by 20 or fewer wasn't the problem. Lots there, but manageable. It was the vast majority outside that range that caused the problem, so I decided to throw away any number abundant/deficient by more than 500.

i suspect, but didn't want to overemphasize, that the vast majority of integers are of the "unmanageable" flavor. that is to say, there are very few sets outside of my Unusual Sets of the Prime*Perfect form that we can find some generating expression(s) for. nevertheless, that all powers of 2 are deficient-by-1 and no other numbers deficient-by-1 but powers of 2 are known, is good enough reason to plow through the mash & see what we can see. maybe other powers exhibit a similar nature? :shrug:

 

while somewhat trivial, it may be worth mentioning that all primes, p, are deficient by p-1. on the somewhat un-trivial side, when dealing with an infinite field, one should be cautious about disregarding a particular ground without...erhm...proper grounds. :lol: of course, we are damned if we do and damned if we don't look at everything within our sight. :)

 

Taking a look at the data collected so far, I've added abundant-by-56 to the file collecting abundant-by-12. As post#28 in this thread shows, numbers of the form 6*p are abundant by 12. This is clearly because 6 is a perfect number. The same logic applies to the other perfect numbers, so I'm segregating 56 (2x28) as well.

 

i named the abundant-by-56 "Bizarre Numbers" when i found them, and have given the following names for the sets generated by the next 4 Perfect Numbers.

abundant-by-992 (Perfect 496*Prime) - Peculiar Numbers

abundant-by-16256 (Perfect 1828*Prime) - Curious Numbers

abundant-by-(Perfect 33550336*Prime) - Quirky Numbers

abundant-by-17179738112 (Perfect 8589869056*Prime) - Freakish Numbers

 

 

Part of the programme involves creating an array of all the factors, so I wanted an idea of how large that array would have to be. I set up a fourth output file containing all numbers with more than 300 factors. So far, the largest I've found has 719 factors (up to around 75 million). Only there's not just one of them. The list has three:

...

Can anyone figure out why the number of factors clusters like that? 3937 have exactly 319 factors, 3277 numbers have 383 ... very odd, but we like odd, don't we! :hihi:

 

indeed we do like odd numbery things! you may find this dusty old thread of interest as it pertains to your observations/questions. >> Big "R" & The Hunt For Phat Numbers

 

that's all from the turtle's shell for now. thnx for your interests & participations fellas!!! :)

[Donk]

A preliminary finding...

 

I thought that concentrating on plus/minus 20 would give a manageable data set. It's certainly done that! Only 193 hits in 180 million numbers. It would have been a LOT more if I'd left the +12 values in, of course.

[Qfwfq]

Can anyone figure out why the number of factors clusters like that? 3937 have exactly 319 factors, 3277 numbers have 383 ...

I can't figure out your listings very well, do you mean the number of divisors or the number of prime factors?

 

Modest appears to have taken it with the first meaning but I find it odd, in this case, that 319 and 383 would be frequent. :shrug:

[Turtle]

A preliminary finding...

 

I thought that concentrating on plus/minus 20 would give a manageable data set. It's certainly done that! Only 193 hits in 180 million numbers. It would have been a LOT more if I'd left the +12 values in, of course.

 

strip out the other "unusuals" in that interval & it will be even leaner. (+56, +992, +16256)

 

I can't figure out your listings very well, do you mean the number of divisors or the number of prime factors?

 

Modest appears to have taken it with the first meaning but I find it odd, in this case, that 319 and 383 would be frequent.

 

looking at his listing of the 3 numbers with 719 factors, and counting them, it must be number of divisors. can't we find the number of divisors by finding the number of combinations of the prime factors? anyway, i'd like to see separate lists of those with 319 & 383 factors; then we might get a handle on the why & how of it. while 319 is composite, 383 & 719 are prime. :shrug: :)

 

i better review the phat numbers thread myself as we may have coverred some of this there. :D :doh: :D off to the races then . . . . . . :hihi:

[Donk]

I can't figure out your listings very well, do you mean the number of divisors or the number of prime factors?

 

Modest appears to have taken it with the first meaning but I find it odd, in this case, that 319 and 383 would be frequent. :shrug:

If all the prime factors are single powers, then the number of combinations is determined by the binomial function, and only certain numbers would appear:

 

e.g.:

3 primes > 1+3+3 = 7 factors (including 1 but excluding "all")

4 primes > 1+4+6+4 = 15 factors

5 primes > 1+5+10+10+5 = 31 factors

...

n primes > 2^n-1 factors.

 

So you'd get only certain numbers cropping up, with nothing in between them. Multiple powers change the rules. When the first seven primes are all 2's, that reduces the number of unique combinations. Clearly, though, there's still some sort of order left in it, hence the clustering of values.

 

Does that make any kind of sense? I didn't get any sleep last night, and I'm coming to the end of a long, hard day :hihi:

[Turtle]

... When the first seven primes are all 2's, that reduces the number of unique combinations.

...

Does that make any kind of sense? I didn't get any sleep last night, and I'm coming to the end of a long, hard day :D

 

well, all those 2's add unique combinations, not reduce, because from them you get the multipliers 4, 8, 16, etc., each multiplied by the other prime factors and if they be powers^2 & above, each of those unique multipliers. your own list of the 719's shows this clear enough. :hihi: i think craig is good at the combinatorics we need here. :D

 

get some rest; you've earned it! :) :shrug: