[seqfan] interesting relationship between A008507, A000217 and A033286

[Bob Selcoe]

Hi Seqfans,

This may be old hat, but...

A008507 = number of odd composites < n-th odd prime:  {0, 0, 0, 1, 1, 2, 2,
3, 5, 5, 7, 8...}, offset 1.
A000217 = Triangular numbers: {0, 1, 3, 6, 10, 15, 21, 28, 36...},  offset
0.
A033286 = n*prime(n): {2, 6, 15, 28, 55, 78, 119...},  offset 1.

It appears that for n>=4, A008507(n) = k+1, where A000217(j) is the smallest
triangular number such that A000217(j) - A033286(n+1) also is a triangular
number, i.e., A000217(k).

Example n=29, A008507(29) = 27:

A033286(30) = 3390,  A000217(86) = 3741.
3741-3390 = 351 = A000217(26);  k=26, 26+1 = 27.

This seems counterintuitive to me.  Is there a proof (or alternatively, a
counter-example)?  I can't program, so would someone be willing to check if
this holds for fairly large n?

I've posted the conjecture on A008507 (not published as of this writing) so
please feel free to edit/comment there.

Cheers,
Bob Selcoe