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%I #14 Jan 06 2013 15:03:19
%S 1,2,2,2,2,2,2,3,1,2,3,2,4,0,2,2,3,4,1,3,1,3,3,3,2,3,2,3,2,2,4,2,7,1,
%T 3,2,1,6,4,4,3,1,3,2,3,6,3,6,0,3,3,2,6,2,4,1,3,4,3,3,4,4,1,1,1,3,3,6,
%U 2,2,2,2,7,1,3,3,2,5,2,5,2,1,5,1,4,1,4,4,1,3,2,3,4,2,3,4,2,5,1,3
%N Number of ways to write n=x+y (x,y>=0) with {6x-1,6x+1} a twin prime pair and y a triangular number
%C Conjecture: a(n)>0 for all n>48624 not equal to 76106.
%C We have verified this for n up to 2*10^8. It seems that 723662 is the unique n>76106 which really needs y=0 in the described representation.
%C Compare the conjecture with another Sun's conjecture associated with A132399.
%D Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), no. 1, 65-76.
%H Zhi-Wei Sun, <a href="/A187785/b187785.txt">Table of n, a(n) for n = 1..100000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e a(9)=1 since 9=3+3(3+1)/2 with 6*3-1 and 6*3+1 both prime.
%t a[n_]:=a[n]=Sum[If[PrimeQ[6(n-k(k+1)/2)-1]==True&&PrimeQ[6(n-k(k+1)/2)+1]==True,1,0],{k,0,(Sqrt[8n+1]-1)/2}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A001097, A000217, A132399, A187759, A187757, A219157.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jan 06 2013
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