[Turtle]

Set of Non-Figurate Numbers: {7,8,11,13,14,17,19,20,23,26,29,31,32,37,38,41,43...}


, for S > 2 and n > 2


i stumbled on this set while doing some mappings of figurate numbers, F_s,n. i have defined as Non-Figurate any integer that is not a solution to the above generalized equation for figurate numbers, for integer variable values s>2 & n> 2. (i have looked, but failed, to find any mention of this set, so if it is known and/or has a different name, somebody stop me. )


the serendipitous find of those non-figurate elements listed above, came by viewing a verbose list of all solutions F_s,n to s=10 & n=20 and noticing the elements as missing. what i want to do now is extend the list for the purpose of having a larger population of non-figurate numbers to analyze, scrutinize, vocabularize, and generally poke a pure math stick into their innards.


certainly one approach is a program that mimics my method, i.e. finding all figurate numbers up to a certain value using the equation and then looking to see what integers are absent. my programming skills and resources are sorely lacking to progress this on my own any further than i have, and so part of my appeal here is to programmers for help.


another course might be to go at it algebraically, though this has the scent of diophantine jugglings if i am not mistaken. not exactly my expertise either.


Note: somewhere here at hypog we once had a discussion about the order of operations and parentheses in the generalized equation as i wrote it. alas i can't find that discussion. i found the equation as i wrote it in another thread and then just added math tags to it here. it may not be correct. we can't have that, so again i need help. here's a specific set of s & n & the expected result: S=3 & n=3, (n/2)*((s-2)*n)-s+4) = 6

if that Latex representation above has an order of operations error that gives the wrong anwser, or if their is a more clear way to represent it, please pop in & correct me. don't want to just trust this one to my own judgement.


that's it for now. ....................

******************* Conclusions Summary Index *************************
under construction...


proposition:
proposer:
proof:
proofer:
posteriors:
posteriorer:

*************************
proposition: there exists a set of natural numbers that do not have [integer] solutions [n,s],n={3,4,5,...}, s={3,4,5,...} to the generalized equation for polygonal numbers P=1/2(n²s-2n²-ns+4n)
proposer: turtle [ post #1 ]
proof: observation of verbose list of polygonal numbers
proofer: turtle [ post #1 ]
posteriors: defined set denominated "non-polygonal numbers"
posteriorer: turtle
posteriors: extended listings of non-polygonal set elements
posteriorers: modest, jay-q, donk, phillip

******************************
proposition: all prime numbers belong to the set of non-polygonal numbers
proposer: turtle [ post #12 ]
proof: yes
proofer: Kharakov [ post #307 & #309 ]
proofer: qfwfq [ post #314 ]
posteriors: a fundamental conclusion on the set of non-polygonals.
posteriorers: concensus

**************************
proposition: find a method to determine if a natural number is polygonal and if so in how many ways.
proposer: diophantus
proof: yes. method given
proofer: donk [ post #32 ]
posteriors: a fundamental method
posteriorer: concensus

****************************
proposition: all prime powers of 2 belong to the set of non-polygonal numbers.
proposer: turtle [ post #4 ]
proof: none
proofer: none
posteriors: the propositon is confirmed by inspection to 2²⁷⁹
posteriorer: modest [ post# 61 ]

****************************
proposition: no multiples of 3 belong to the set of non-polygonal numbers.
proposer: turtle [ post #314

turtle said:

Posted 02 November 2006 - 02:33 PM... Following this line in view of the goal of using the generalized expression of figurate numbers, (n/2)*((s-2)*n)-s+4) when S >= 3 and n >= 3, to find non-figurate numbers, we can by analysis
of the Katabatak table for figurate numbers eliminate a good number of possibilities for n. On the assumption that no non-figurate numbers have K transforms of 3 , 6 , or 9 (in base ten), then from the table we see three columns have only these transforms, and because the pattern is infinitely repeated over the set of integers we can conjecture that no indices in n of the forms (9m-6), (9m-3), or (9m) need trying.
This is in retrospect tantamount to saying that any number divisible by 3 is figurate.

proposer: donk [ post #100 ]

donk said:

Posted 23 December 2009 - 11:04 PM... Donk's proposition 3.1 on non-figurate numbers:
No non-figurate number is divisible by 3.

proof: yes.
proofer: turtle [ post #103 ]
posteriors: a fundamental conclusion on the set of non-polygonal numbers
posteriorer: concensus

**********************
proposition: any non-polygonal number divisible by 5 is divisible by 10.
proposer: donk [ post #100 ]
proof: yes
proofer: searching...
posteriors:
posteriorer:

*************************
proposition: all even non-polygonal numbers have the form 6n+2
proposer: idmclean [ post # 199 ]
proof: yes
proofer: turtle [ post #? ]
posteriors:
posteriorer:

** through post #338

[Jay-qu]

Firstly I assumed that you meant:

(removed extraneous parenthesis)

and used this code:

s=n=3
F=[]
while s<1000:
    while n<1000:
        F.append((n/2)*((s-2)*n-s+4))
        n+=1
    n=3
    s+=1

to generate the first ~10^5 Figurate numbers.

Then we just have to take the complement of this set. I wasnt sure if python had a built in function for this so I wrote this extra bit of code (brute force approach :D )

nF=[]
i=6
j=0
maximum=max(F)
length=len(F)
isNonFigurate=True
while i<maximum:
    while (j<length):
        if i==F[j]:
            isNonFigurate=False
        j+=1
    if isNonFigurate:
        nF.append(i)
    else:
        isNonFigurate=True
    j=0
    i+=1

This produced a set that is a bit different to yours..

so you say s=3, n=2 should yield a 6? I dont see how you can get that, please fix up your bracketing so I can fix my code :)

over

[Turtle]

Jay-qu said:

Firstly I assumed that you meant:

This produced a set that is a bit different to yours..
...
so you say s=3, n=2 should yield a 6? I dont see how you can get that, please fix up your bracketing so I can fix my code :)

over

roger. i suspected i was being myopic on that extra parenthesis. thnx! :bow: now onward to claim our booty! :piratesword:

i gave n=3, not n=2 in my example.
s=3, n=3 ( this gives us the 3rd 3-sided (or triangular) number which is 6)









i think we're back on track now. :thumbs_up :steering:

[modest]

Turtle said:

I think you're still missing a necessary parentheses It has one more right parentheses than left. I believe you mean to convey this:



since multiplication comes before addition without the need for parentheses you could omit a pair:



I don't see an obvious way to simplify it any further. Maybe:



~modest

[Turtle]

modest said:

I think you're still missing a necessary parentheses It has one more right parentheses than left. I believe you mean to convey this:



since multiplication comes before addition without the need for parentheses you could omit a pair:



I don't see an obvious way to simplify it any further. Maybe:



~modest

:thumbs_up i agree. i may have carried that expression form and those extra parentheses over from the original Basic program i wrote to generate the lists. previous to that i did it longhand. once i got something giving the correct output, i let it be. :lol: that one parenthesis is missing here is a typo. :doh: one of my programming difficulties alluded to in the op is my antique Basic not allowing integer values much over 2 billion.

ps do you see any means of using the equation as you re-wrote it last there, to find non-figurate numbers without a search & comparison?

[modest]

Turtle said:

ps do you see any means of using the equation as you re-wrote it last there, to find non-figurate numbers without a search & comparison?

I don't think so. I'm pretty ridiculously bad at that sort of thing though.

I think this code would work for a brute force kind of search:

for($checknum = 6; $checknum < 1000; $checknum++){
  $found = 0;
  for($n = 2; $n < $checknum and $found == 0; $n++) {
    for($s = 2; $s < $checknum and $found == 0; $s++) {
    if (($n/2)*(($s-2)*$n-$s+4) == $checknum) {$found = 1;}
    }
  }
  if ($found == 0) {print "$checknum, ";}
}

F < 1000 =

7,8,11,13,14,17,19,20,23,26,29,31,32,37,38,41,43,44,47,50,53
,56,59,61,62,67,68,71,73,74,77,79,80,83,86,89,97,98,101,103,
104,107,109,110,113,116,119,122,127,128,131,134,137,139,140,143,
146,149,151,152,157,158,161,163,164,167,170,173,179,181,182,187,
188,191,193,194,197,199,200,203,206,209,211,212,218,221,223,224,
227,229,230,233,236,239,241,242,248,251,254,257,263,266,269,271,
272,277,278,281,283,284,290,293,296,299,302,307,308,311,313,314,
317,319,320,323,326,329,331,332,337,338,347,349,350,353,356,359,
362,367,368,371,373,374,377,379,380,383,386,389,391,392,397,398,
401,404,407,409,410,413,416,419,421,422,431,433,434,437,439,440,
443,446,449,452,457,458,461,463,464,467,470,473,476,479,482,487,
488,491,493,494,497,499,500,503,509,517,518,521,523,524,527,530,
533,536,539,541,542,547,548,551,554,557,563,566,569,571,572,577,
578,581,583,584,587,589,593,599,601,602,607,608,611,613,614,617,
619,620,623,626,629,631,632,638,641,643,644,647,649,650,653,656,
659,661,662,667,668,673,674,677,683,686,689,691,692,698,701,704,
707,709,710,713,716,719,722,727,728,731,733,734,737,739,740,743,
746,749,751,752,757,758,761,767,769,770,773,776,779,787,788,791,
794,797,799,800,803,806,809,811,812,817,818,821,823,824,827,829,
830,839,842,851,853,854,857,859,860,863,866,869,872,877,878,881,
883,884,887,890,893,896,899,901,902,907,908,911,913,914,917,919,
920,923,926,929,937,938,941,943,944,947,950,953,956,959,962,967,
968,971,974,977,979,980,983,986,989,991,992,997,998,

Do those look correct?

I'll try to figure out a sequence or function.

~modest

[Jay-qu]

Ok I slipped, but its fixed now and I get this set of numbers which look like Modest's

7 , 8 , 11 , 13 , 14 , 17 , 19 , 20 , 23 , 26 , 29 , 31 , 32 , 37 , 38 , 41 , 43 , 44 , 47 , 50 , 53 , 56
, 59 , 61 , 62 , 67 , 68 , 71 , 73 , 74 , 77 , 79 , 80 , 83 , 86 , 89 , 97 , 98 , 101 , 103 , 104 , 107 , 109 , 
110 , 113 , 116 , 119 , 122 , 127 , 128 , 131 , 134 , 137 , 139 , 140 , 143 , 146 , 149 , 151 , 152 , 157 ,
 158 , 161 , 163 , 164 , 167 , 170 , 173 , 179 , 181 , 182 , 187 , 188 , 191 , 193 
, 194 , 197 , 199 ,

So what type of analysis do you want for this turtle?

[Turtle]

modest said:

Trickle said:

ps do you see any means of using the equation as you re-wrote it last there, to find non-figurate numbers without a search & comparison?

I don't think so. I'm pretty ridiculously bad at that sort of thing though.

roger. do you agree with my assessment that it is/would/could be a matter of diophantine equations to go at it as i suggested?

Modernt said:

I think this code would work for a brute force kind of search:

for($checknum = 6; $checknum < 1000; $checknum++){
  $found = 0;
  for($n = 2; $n < $checknum and $found == 0; $n++) {
    for($s = 2; $s < $checknum and $found == 0; $s++) {
    if (($n/2)*(($s-2)*$n-$s+4) == $checknum) {$found = 1;}
    }
  }
  if ($found == 0) {print "$checknum, ";}
}

F < 1000 =

7,8,11,13,14,17,19,20,23,26,29,31,32,37,38,41,43,44,47,50,53
,56,59,61,62,67,68,71,73,74,77,79,80,83,86,89,97,98,101,103,
104,107,109,110,113,116,119,122,127,128,131,134,137,139,140,143,
146,149,151,152,157,158,161,163,164,167,170,173,179,181,182,187,
188,191,193,194,197,199,200,203,206,209,211,212,218,221,223,224,
227,229,230,233,236,239,241,242,248,251,254,257,263,266,269,271,
272,277,278,281,283,284,290,293,296,299,302,307,308,311,313,314,
317,319,320,323,326,329,331,332,337,338,347,349,350,353,356,359,
362,367,368,371,373,374,377,379,380,383,386,389,391,392,397,398,
401,404,407,409,410,413,416,419,421,422,431,433,434,437,439,440,
443,446,449,452,457,458,461,463,464,467,470,473,476,479,482,487,
488,491,493,494,497,499,500,503,509,517,518,521,523,524,527,530,
533,536,539,541,542,547,548,551,554,557,563,566,569,571,572,577,
578,581,583,584,587,589,593,599,601,602,607,608,611,613,614,617,
619,620,623,626,629,631,632,638,641,643,644,647,649,650,653,656,
659,661,662,667,668,673,674,677,683,686,689,691,692,698,701,704,
707,709,710,713,716,719,722,727,728,731,733,734,737,739,740,743,
746,749,751,752,757,758,761,767,769,770,773,776,779,787,788,791,
794,797,799,800,803,806,809,811,812,817,818,821,823,824,827,829,
830,839,842,851,853,854,857,859,860,863,866,869,872,877,878,881,
883,884,887,890,893,896,899,901,902,907,908,911,913,914,917,919,
920,923,926,929,937,938,941,943,944,947,950,953,956,959,962,967,
968,971,974,977,979,980,983,986,989,991,992,997,998,

Do those look correct?

I'll try to figure out a sequence or function.

~modest

yes correct!

J-quenator said:

Ok I slipped, but its fixed now and I get this set of numbers which look like Modest's.

yes doubly correct!!

now my only caveat is that by correct i mean matches my list done by hand exactly and then presuming that is the proof of the accuracy of your programs/algorithms. as we are plowing new ground, this is as good as it gets!!

muchas gracias mi amigos!! i'll be gazing at the list as modest gave it (more elements is all) to see what eye can see. already i see it is far more dense than i suspected.

[modest]

Turtle said:

roger. do you agree with my assessment that it is/would/could be a matter of diophantine equations to go at it as i suggested?

Dio-phan-a-what now? Uh... You got me. I'm sure Jay would know better.

I did have a crack at the problem. I thought I was being terribly clever—starting out with this data set:

6, 9, 12, 15, 18, 21, 24, 27, 30, 33,
10, 16, 22, 28, 34, 40, 46, 52, 58, 64,
15, 25, 35, 45, 55, 65, 75, 85, 95, 105,
21, 36, 51, 66, 81, 96, 111, 126, 141, 156

It seemed that the numbers running left/right follow a fairly predictable pattern as do the numbers arranged in columns. Introducing to get rid of the rows I got:

6+n(6-3)
10+n(10-4)
15+n(15-5)
21+n(28-6)

The series 3,4,5,6 is clearly the next variable and the other numbers can be related to that number by dividing it by two, adding one half, then multiplying it by the original number. So, I got:

[(3•0.5+0.5)3]+n([(3•0.5+0.5)3]-3)
[(4•0.5+0.5)4]+n([(4•0.5+0.5)4]-4)
[(5•0.5+0.5)5]+n([(5•0.5+0.5)5]-5)
[(6•0.5+0.5)6]+n([(6•0.5+0.5)6]-6)

Replacing 3,4,5,6... with gives:



or, rearranged:



where and

Only after doing this did I realize that all I've done is to create an expression which generates figurate numbers greater than 5. Since your function does that just fine I've accomplished absolutely nothing :doh:

~modest

[Turtle]

modest said:

Dio-phan-a-what now? Uh... You got me. I'm sure Jay would know better.

so uh..erh...that's what she said. mind you i have only a passing familiarity. nonetheless, that's enough to start. can't seem to get the wolfram page on the subject to come up so will start with you-know-who. >> Diophantine equation - Wikipedia, the free encyclopedia

wicked pedeac said:

...Diophantine analysis

Typical questions
The questions asked in Diophantine analysis include:

1.Are there any solutions?
2.Are there any solutions beyond some that are easily found by inspection?
3.Are there finitely or infinitely many solutions?
4.Can all solutions be found, in theory?
5.Can one in practice compute a full list of solutions?
These traditional problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles.

so what i meant to imply, given our generalized expression/equation for figurate numbers, and given that we know by demonstration/inspection that there exist integers not having solutions under the equation when the variables are restricted to integers, is there a method to use that generalized equation to compute a full list of solutions, ie., my non-figurate numbers.

Modestein said:

I did have a crack at the problem. I thought I was being terribly clever—starting out with this data set:
...
Only after doing this did I realize that all I've done is to create an expression which generates figurate numbers greater than 5. Since your function does that just fine I've accomplished absolutely nothing

~modest

indeed. so too did i go down a similar roundybout path. i tried looking for a pattern of even/odd but to no avail. not that it's fo shizzle not there, but if it is, then it is very long. still, i better show what i did & how it apparently got us little if nothing.

i broke the list into groups of 100's, then i counted the elements in each group for the first few groups and that count is in brackets at the end of those rows. this is to get a grip on my earlier expressed surprise at how dense the set is with, so far as we have gone, over 1/3 of all integers being non-figurate.
the E's & O's at the bottom are a map of even/odd following the order of the set.

8,11,13,14,17,19,20,23,26,29,31,32,37,38,41,43,44,47,50,53,56,59,61,62,67,68,71,73,74,77,79,80,83,86,89,97,98,[37]

101,103,104,107,109,110,113,116,119,122,127,128,131,134,137,139,140,143,146,149,151,152,157,158,161,163,164,167,170,173,179,181,182,187,188,191,193,194,197,199,[40]

200,203,206,209,211,212,218,221,223,224,227,229,230,233,236,239,241,242,248,251,254,257,263,266,269,271,272,277,278,281,283,284,290,293,296,299,[36]

302,307,308,311,313,314,317,319,320,323,326,329,331,332,337,338,347,349,350,353,356,359,362,367,368,371,373,374,377,379,380,383,386,389,391,392,397,398,[38]

401,404,407,409,410,413,416,419,421,422,431,433,434,437,439,440,443,446,449,452,457,458,461,463,464,467,470,473,476,479,482,487,488,491,493,494,497,499, [38]

500,503,509,517,518,521,523,524,527,530,533,536,539,541,542,547,548,551,554,557,563,566,569,571,572,577,578,581,583,584,587,589,593,599,[34]

601,602,607,608,611,613,614,617,619,620,623,626,629,631,632,638,641,643,644,647,649,650,653,656,659,661,662,667,668,673,674,677,683,686,689,691,692,698,

701,704,707,709,710,713,716,719,722,727,728,731,733,734,737,739,740,743,746,749,751,752,757,758,761,767,769,770,773,776,779,787,788,791,794,797,799,

800,803,806,809,811,812,817,818,821,823,824,827,829,830,839,842,851,853,854,857,859,860,863,866,869,872,877,878,881,883,884,887,890,893,896,899,

901,902,907,908,911,913,914,917,919,920,923,926,929,937,938,941,943,944,947,950,953,956,959,962,967,968,971,974,977,979,980,983,986,989,991,992,997,998

E O O E O O E O E O O E O E O O E O E O E O O E O E O O E O O E O E O O E O O E O O E O E O E O E O E O O E O E O O E O E O O E O E O O O E O E O O E O O E 

off hand reviewing the list, the largest interval between pairs i see is 9, as in {...572,577...} or {...830,839...}. now if we.......... . . . . . . . . .

[modest]

Turtle said:

off hand reviewing the list, the largest interval between pairs i see is 9, as in {...572,577...} or {...830,839...}. now if we.......... . . . . . . . . .

Interesting. Highlighting non-figs on a number chart arranged by 9's looks patternish:



I'll arrange them by 10's and edit it on to the post when done...

~modest

EDIT:

By 10's:



By 6's:

[Turtle]

modest said:

Interesting. Highlighting non-figs on a number chart arranged by 9's looks patternish:

I'll arrange them by 10's and edit it on to the post when done...

~modest

EDIT:

By 10's:

By 6's:

interesting. very katabatakesque. all that remains is to carry on.

i just noticed that the non-figurate set contains all the primes >7! well, as far as i looked before feeling ready to jump to such a conclusion anyway. nonetheless, nothing leaped, nothing lopped i always say. moreover, the set contains composite odd numbers as well. that stickies our wickets a bit. fascinating.

[modest]

Turtle said:

i just noticed that the non-figurate set contains all the primes >7!

I had noticed that. It also looked like the only non-figs in the set {1, 4, 7, 10, 13...} were prime {7, 13, 19, 31, 37, 43, 61... } But, 187 breaks that trend. It is non-prime and in the counting-by-threes set.

~modest

[Turtle]

modest said:

I had noticed that. It also looked like the only non-figs in the set {1, 4, 7, 10, 13...} were prime {7, 13, 19, 31, 37, 43, 61... } But, 187 breaks that trend. It is non-prime and in the counting-by-threes set.

~modest

what is the set {1, 4, 7, 10, 13...} from?? 1,4,7 & 10 are all figurate, not non-figurate. ?

if we presume it's true that the set of primes > 7 are a proper subset of non-figurate numbers, then we can subtract them and try concentrating on non-prime non-figurate numbers to find a pattern. inasmuch as there is no known equation that gives all primes, looking for one to give all non-figurate is a complication on that problem. curiouser & more curiouser.

that's it; i'm out. . . . . . . . . . . . .

[modest]

Turtle said:

what is the set {1, 4, 7, 10, 13...} from??

Counting by three. x = x + 3.

Turtle said:

1,4,7 & 10 are all figurate, not non-figurate. ?

Yes. It looked as if the only non-figurate numbers in the counting by three set were prime. They are {7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151...}. Those are the only non-figurate numbers in the counting-by-three set and they all appear to be prime. But the number 187 breaks that trend. It is non-figurate, in the counting by three set, but not a prime.

If we're going to find a function or expression that gives non-figurates, I think we need to unscramble the linear sequence we have and arrange it in the way it would be output from the function. For example, I was able to find the function for this data:

6, 9, 12, 15, 18, 21, 24, 27, 30, 33...
10, 16, 22, 28, 34, 40, 46, 52, 58, 64...
15, 25, 35, 45, 55, 65, 75, 85, 95, 105...
21, 36, 51, 66, 81, 96, 111, 126, 141, 156...

because each row and each column do something predictable. That's the only way I really know how to do something like this—especially since it's clearly going to have more than one variable. But, just when I think I find the beginnings of a useful layout, it breaks. The closest I got was starting at two and counting by 12:

{2, 14, 26, 38, 50, 62, 74, 86, 98, 110, 122, 134, 146, 158, 170, 182...}

Those are all non-figurate and in a predictable pattern which looks like it might be the top row in a useful arrangement. But, the sequence breaks at 506 which is in the counting-by-12 set but it's not a non-figurate.

If there is a function it may be beyond such a simple method.

~modest