A263998 Number of ordered ways to write n as x^2 + 2*y^2 + p*(p+d)/2, where x and y are nonnegative integers, d is 1 or -1, and p is prime.

1, 1, 3, 3, 3, 3, 4, 1, 4, 4, 3, 7, 2, 4, 5, 2, 3, 4, 7, 3, 7, 5, 4, 5, 5, 3, 5, 8, 3, 8, 3, 4, 6, 5, 4, 5, 10, 2, 11, 4, 2, 6, 3, 6, 3, 7, 5, 5, 3, 3, 6, 5, 6, 8, 7, 3, 9, 5, 4, 9, 5, 4, 4, 8, 4, 5, 8, 2, 11, 5, 5, 9, 5, 6, 8, 6, 5, 10, 8, 3, 4, 13, 4, 10, 7, 4, 12, 6, 7, 4, 10, 6, 7, 6, 4, 9, 5, 5, 8, 11

Offset

1,3

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 8.

This is similar to the conjecture in A262785.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Some mysterious representations of integers, a message to Number Theory Mailing List, Oct. 25, 2015.

Example

 a(1) = 1 since 1 = 0^2 + 2*0^2 + 2*(2-1)/2 with 2 prime.
a(2) = 1 since 2 = 1^2 + 2*0^2 + 2*(2-1)/2 with 2 prime.
a(8) = 1 since 8 = 0^2 + 2*1^2 + 3*(3+1)/2 with 3 prime.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
f[d_, n_]:=f[d, n]=Prime[n](Prime[n]+(-1)^d)/2
Do[r=0; Do[If[SQ[n-f[d, k]-2x^2], r=r+1], {d, 0, 1}, {k, 1, PrimePi[(Sqrt[8n+1]-(-1)^d)/2]}, {x, 0, Sqrt[(n-f[d, k])/2]}]; Print[n, " ", r]; Continue, {n, 1, 100}]

Crossrefs

Cf. A000040, A000217, A000290, A262311, A262785, A263992.

Sequence in context: A264097 A177013 A092282 * A048181 A091799 A276863

Adjacent sequences: A263995 A263996 A263997 * A263999 A264000 A264001

Keyword

nonn

Author

Zhi-Wei Sun, Oct 31 2015

Status

approved