# [seqfan] Re: A (new) problem

Charles Greathouse charles.greathouse at case.edu
Sun Apr 25 18:40:15 CEST 2010

```Interesting question.  Essentially: is there some k for which all
positive integers can be written as the sum of an a-polygonal and a
b-polygonal number, for 3 <= a <= b <= k?

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Sun, Apr 25, 2010 at 9:59 AM, Vladimir Shevelev <shevelev at bgu.ac.il> wrote:
> Dear seqfans,
>
> I propose you the following problem (maybe, earlier somebody heard about that, I did not).
>
> It is well known that every nonnegative integer is either triangular number or sum of 2 triangular numbers or sum of 3 triangular numbers. Starting with set of the first 3 triangular numbers {0,1,3}, we see that every nonnegative integer up to 4 is a sum of 2 triangular numbers, but 5 is not. With this moment we consider union of  triangular numbers and squares (A054686). Further, we see that every nonnegative integer up to 22 is a sum of 2 terms of A054686, but 23 is not. With this moment we consider union of A054686 and sequence of pentagonal numbers (A000326) and see  that every  nonnegative integer up to 61 is a sum of 2 terms of this union, but 62 is not. With this moment  we join sequence of hexagonal numbers (A000384) etc. Thus we obtain sequence 5,23,62,...
> The problem is to prove or disprove that this sequence is infinite. I beleive that this sequence is, indeed, infinite.
>
> Best regards,