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%I #37 Oct 15 2013 09:43:03
%S 1,1,1,3,1,1,3,2,2,2,3,2,3,3,1,2,3,3,3,2,2,5,3,2,2,4,2,2,4,2,2,2,3,1,
%T 3,2,3,2,2,4,3,1,3,5,2,3,4,5,2,4,2,3,3,2,3,5,4,2,4,1,4,3,5,4,3,5,3,4,
%U 3,3,6,4,2,5,4,3,4,5,5,2,4,4,2,3,6,4,2,3,5,4,3,5,1,4,3,6,3,5,7,3
%N Number of ordered ways to write n = x*(x+1)/2 + y with y*(y+1)/2 + 1 prime, where x and y are nonnegative integers.
%C Conjecture: a(n) > 0 for all n > 0. Moreover, if n > 0 is not among 1, 3, 60, then there are positive integers x and y with x*(x+1)/2 + y = n such that y*(y+1)/2 + 1 is prime.
%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="/A229166/b229166.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.
%e a(6) = 1 since 6 = 2*3/2 + 3 with 3*4/2 + 1 = 7 prime.
%e a(60) = 1 since 60 = 0*1/2 + 60 with 60*61/2 + 1 = 1831 prime.
%t T[n_]:=n(n+1)/2
%t a[n_]:=Sum[If[PrimeQ[T[n-T[i]]+1],1,0],{i,0,(Sqrt[8n+1]-1)/2}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000217, A132399, A208244, A230121, A230252.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Oct 15 2013
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