# [seqfan] Re: A (new) problem

franktaw at netscape.net franktaw at netscape.net
Mon Apr 26 01:02:41 CEST 2010

```This relates to a question I have wondered about for some time. A126684
is a sequence with a(n) = O(n^2) such that every non-negative integer
can be represented as the sum of two members of the sequence. However,
it is "irregularly" O(n^2); that is, it is not the case that a(n) = c
n^2 + o(n^2) for some c. I have wondered if there is a sequence that
has a(n) = c n^2 + o(n^2) that has the sum of two members property.

If Vladimir's sequence is finite, then the associated sum of polygonal
numbers provides the example I have been looking for.

-----Original Message-----
From: Vladimir Shevelev <shevelev at bgu.ac.il>

Dear seqfans,

I propose you the following problem (maybe, earlier somebody heard
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,