A232186 Number of ways to write n = p + q (q > 0) with p and p^3 + n*q^2 both prime.

0, 0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 2, 2, 5, 1, 1, 3, 1, 5, 4, 2, 3, 3, 1, 2, 3, 2, 4, 6, 2, 3, 5, 2, 3, 3, 3, 2, 3, 4, 2, 4, 3, 2, 2, 3, 2, 6, 2, 3, 3, 5, 4, 4, 4, 5, 9, 1, 4, 7, 3, 4, 6, 3, 5, 8, 3, 5, 6, 5, 5, 13, 2, 4, 5, 4, 4, 7, 5, 5, 13, 3, 5, 8, 6, 4, 6, 4, 3, 8, 3, 4, 9, 1, 4, 11, 3

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 2.

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.

Example

a(10) = 1 since 10 = 7 + 3 with 7 and 7^3 + 10*3^2 = 433 both prime.
a(11) = 1 since 11 = 5 + 6 with 5 and 5^3 + 11*6^2 = 521 both prime.
a(124) = 1 since 124 = 19 + 105 with 19 and 19^3 + 124*105^2 = 1373959 both prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[Prime[k]^3+n*(n-Prime[k])^2], 1, 0], {k, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A219864, A220413, A232174.

Sequence in context: A284580 A227349 A246028 * A340061 A325757 A161161

Adjacent sequences: A232183 A232184 A232185 * A232187 A232188 A232189

Keyword

nonn

Author

Zhi-Wei Sun, Nov 20 2013

Status

approved