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%I #13 Nov 10 2013 02:56:15
%S 0,1,1,2,2,1,3,2,2,3,2,4,4,2,2,3,6,1,5,2,3,4,3,5,1,6,4,5,2,5,8,5,6,5,
%T 3,6,10,5,5,9,8,6,13,3,5,12,9,6,4,6,7,18,5,7,4,7,14,6,11,7,16,6,7,13,
%U 6,9,13,8,6,11,7,15,14,6,11,11,6,15,12,9,6,20,9,5,20,9,8,14,15,8,9,18,7,15,6,16,17,9,10,7
%N Number of ways to write n = x + y (x, y > 0) with x*(x+1)/2 + y^2 prime.
%C Conjecture: a(n) > 0 for all n > 1.
%C This implies that there are infinitely many primes of the form x*(x+1)/2 + y^2 (i.e., the sequence A228424 has infinitely many terms).
%C For m = 3, 4, 5, ... the m-gonal numbers are given by p_m(x) = (m-2)*x*(x-1)/2 + x (x = 0, 1, 2, ...). We note that there are many pairs m > k > 2 such that all sufficiently large integers n can be written as x + y (x, y > 0) with p_k(x) + p_m(y) prime. For example, we conjecture that the pair (k, m) works if k is among 3, 4, 6 , and m > k is not congruent to k modulo 2. For k = 5, we guess that the pair (5, m) works if m is congruent to 0 or 4 modulo 6.
%C We conjecture that the only pairs (k,m) with 2 < k <= 10 and k< m <= 100 such that any integer n > 1 can be written as x + y (x, y > 0) with p_k(x) + p_m(y) prime, are as follows: (3,4),(3,6),(3,28),(3,46),(3,52),(3,82),(3,88),(4,7),(4,15),(4,25),(4,27),(4,37),(4,43),(4,63),(4,67),(4,97),(6,25),(6,43),(6,73),(7,10),(7,18),(7,100),(10,15),(10,19),(10,27),(10,37),(10,55),(10,75),(10,79),(10,87),(10,99).
%C We also conjecture that any integer n > 1 can be written as x + y (x, y > 0) with p_k(x) + p_{k+1}(y) prime, if and only if k is among 3, 39, 99.
%H Zhi-Wei Sun, <a href="/A228425/b228425.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 + 4 with 2*3/2 + 4^2 = 19 prime.
%e a(18) = 1 since 18 = 7 + 11 with 7*8/2 + 11^2 = 149 prime.
%e a(25) = 1 since 25 = 1 + 24 with 1*2/2 + 24^2 = 577 prime.
%t a[n_]:=Sum[If[PrimeQ[x(x+1)/2+(n-x)^2],1,0],{x,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000217, A000290, A228424.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Nov 10 2013
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