A199800 Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime

0, 0, 1, 2, 2, 2, 2, 3, 2, 3, 0, 4, 2, 4, 3, 2, 2, 3, 3, 5, 3, 3, 3, 4, 4, 3, 2, 4, 3, 5, 3, 4, 3, 5, 5, 6, 3, 4, 3, 5, 5, 5, 6, 5, 4, 5, 5, 6, 7, 5, 4, 5, 4, 7, 6, 4, 4, 4, 5, 6, 6, 5, 6, 7, 4, 5, 2, 4, 7, 5, 7, 4, 5, 6, 7, 7, 7, 5, 6, 4, 7, 4, 7, 7, 6, 5, 3, 5, 8, 7, 7, 5, 5, 6, 4, 5, 4, 5, 8, 7

Offset

1,4

Comments

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

This implies the twin prime conjecture, and it has been verified for n up to 10^9.

Zhi-Wei Sun also made some similar conjectures, for example, any integer n>5 can be written as p+q with p, 2q-3 and 2q+3 all prime, and each integer n>4 can be written as p+q with p, 3q-2+(n mod 2) and 3q+2-(n mod 2) all prime.

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(3)=1 since 3=2+1 with 2, 6*1-1 and 6*1+1 all prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[n-k]==True&&PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True, 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]

Crossrefs

Cf. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

Sequence in context: A233417 A299741 A074589 * A338094 A165035 A236531

Adjacent sequences: A199797 A199798 A199799 * A199801 A199802 A199803

Keyword

nonn

Author

Zhi-Wei Sun, Dec 21 2012

Status

approved