Non-Figurate Numbers

[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.

[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 )

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]

Originally Posted by Jay-qu

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! now onward to claim our booty!

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.

[modest]

Originally Posted by Turtle

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]

Originally Posted by modest

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

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. that one parenthesis is missing here is a typo. 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]

Originally Posted by Turtle

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]

Originally Posted by modest

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?

Originally Posted by Modernt

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!

Originally Posted by J-quenator

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]

Originally Posted by Turtle

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

~modest

[Turtle]

Originally Posted by modest

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

Originally Posted by wicked pedeac

...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.

Originally Posted by Modestein

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.......... . . . . . . . . .