[seqfan] A (new) problem

Vladimir Shevelev shevelev at bgu.ac.il
Sun Apr 25 15:59:21 CEST 2010


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, 
Vladimir

 Shevelev Vladimir‎



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