|
%I #9 Nov 10 2013 03:50:40
%S 0,1,1,1,2,1,1,1,3,3,1,2,4,1,3,5,4,3,5,1,4,3,2,1,7,5,1,4,4,4,4,5,2,5,
%T 4,2,6,1,4,9,6,4,6,6,5,6,5,4,6,5,11,9,6,5,10,4,3,9,5,3,11,7,7,10,5,5,
%U 10,5,5,5,4,7,6,6,5,13,7,3,12,5,5,12,6,7,8,6,4,9,7,8,9,10,9,9,8,15,17,9,9,10
%N Number of ways to write n = x + y (x, y > 0) with p(99, x) + p(100, y) prime, where p(m, k) denotes the m-gonal number (m-2)*k*(k-1)/2 + k.
%C Conjecture: a(n) > 0 for all n > 1.
%C See also A228425 for other similar conjectures.
%H Zhi-Wei Sun, <a href="/A228428/b228428.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, preprint, arXiv:0905.0635.
%e a(8) = 1 since 8 = 3 + 5 with p(99, 3) + p (100, 5) = 1279 prime.
%e a(38) = 1 since 38 = 6 + 32 with p(99, 6) + p(100, 32) = 50101 prime.
%t p[m_,x_]:=(m-2)x(x-1)/2+x
%t a[n_]:=Sum[If[PrimeQ[p[99,x]+p[100,n-x]],1,0],{x,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A228425.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Nov 10 2013
|
