cp's OEIS Frontend

A228429 Number of ways to write n = x + y (x, y > 0) with p(39, x) + p(40, y) prime, where p(m, k) denotes the m-gonal number (m-2)*k*(k-1)/2 + k.

%I A228429 #14 Jul 08 2025 07:48:56
%S A228429 0,1,1,1,2,1,1,2,1,1,3,1,3,1,1,4,2,2,4,1,3,1,2,5,5,1,2,3,4,3,11,4,4,2,
%T A228429 3,4,9,6,3,5,6,3,5,4,3,9,6,3,4,7,5,13,8,3,5,5,6,13,9,9,6,3,4,6,7,3,5,
%U A228429 5,8,5,11,8,11,8,5,10,9,5,13,9,10,11,4,7,14,12,6,11,9,5,9,15,5,14,11,6,7,8,13,14
%N A228429 Number of ways to write n =  x + y (x, y > 0) with p(39, x) + p(40, y) prime, where p(m, k) denotes the m-gonal number (m-2)*k*(k-1)/2 + k.
%C A228429 By a conjecture in A228425, we should have a(n) > 0 for all n > 1.
%C A228429  Conjecture: For each m = 3, 4, ..., any sufficiently large integer n can be written as x + y (x, y > 0) with p(m, x) + p(m+1, y) prime.
%H A228429 Zhi-Wei Sun, <a href="/A228429/b228429.txt">Table of n, a(n) for n = 1..10000</a>
%H A228429 Zhi-Wei Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, preprint, arXiv:0905.0635 [math.NT], 2009-2015.
%e A228429 a(9) = 1 since 9 = 5 + 4 with p(39, 5) + p(40, 4) = 607 prime.
%e A228429 a(26) = 1 since 26 = 19 + 7 with p(39, 19) + p (40, 7) = 7151 prime.
%t A228429 p[m_,x_]:=(m-2)x(x-1)/2+x
%t A228429 a[n_]:=Sum[If[PrimeQ[p[39,x]+p[40,n-x]],1,0],{x,1,n-1}]
%t A228429 Table[a[n],{n,1,100}]
%Y A228429 Cf. A000040, A228425, A228428, A228430, A228431.
%K A228429 nonn
%O A228429 1,5
%A A228429 _Zhi-Wei Sun_, Nov 10 2013