A228425 Number of ways to write n = x + y (x, y > 0) with x*(x+1)/2 + y^2 prime.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Nov 10 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1.

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).

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.

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).

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.

Examples

			a(6) = 1 since 6 = 2 + 4 with 2*3/2 + 4^2 = 19 prime.
a(18) = 1 since 18 = 7 + 11 with 7*8/2 + 11^2 = 149 prime.
a(25) = 1 since 25 = 1 + 24 with 1*2/2 + 24^2 = 577 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Crossrefs

Cf. A000040, A000217, A000290, A228424.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[x(x+1)/2+(n-x)^2],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]