A219966 Number of ways to write n=p+q+(n mod 2)q with q<=n/2 and p, q, q+6 all prime

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 2, 2, 3, 2, 3, 3, 2, 1, 4, 3, 1, 4, 3, 1, 4, 2, 3, 3, 2, 2, 4, 3, 2, 4, 2, 2, 5, 3, 4, 5, 2, 1, 5, 3, 2, 4, 1, 1, 5, 4, 4, 4, 3, 2, 5, 3, 2, 4, 3, 4, 5, 3, 4, 6, 3, 3, 6, 3, 3, 8, 5, 2, 6, 3, 4, 6, 2, 2, 9, 5, 3, 5, 4, 2, 6, 4

Offset

1,17

Comments

Conjecture: a(n)>0 for all n>11.

This conjecture is stronger than Goldbach's conjecture and Lemoine's conjecture. It can be further strengthened; see A219055 and the comments there.

Links

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

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(19)=1 since 19=5+2*7 with 5, 7, 7+6 all prime.
a(20)=1 since 20=13+7 with 13, 7, 7+6 all prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[n-(1+Mod[n, 2])Prime[k]]==True, 1, 0], {k, 1, PrimePi[n/2]}]
Do[Print[n, " ", a[n]], {n, 1, 10000}]

Crossrefs

Cf. A219055, A023201, A002375, A046927.

Sequence in context: A271381 A086247 A116452 * A103958 A321696 A321020

Adjacent sequences: A219963 A219964 A219965 * A219967 A219968 A219969

Keyword

nonn

Author

Zhi-Wei Sun, Dec 02 2012

Status

approved