A219025 Number of primes p<n such that 6n-p and 6n+p are both prime

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

Offset

1,11

Comments

Conjecture: a(n)>0 for all n=6,7,...

This has been verified for n up to 10^8.

Zhi-Wei Sun also made the following general conjecture:

Let P(x) be any non-constant integer-valued polynomial with positive leading coefficient. If n is large enough, then there is a prime p<n such that 6P(n)+p and 6P(n)-p are both prime. For example, for P(x)=x(x+1)/2, x^2, x^3, x^4 it suffices to require that n is greater than 1933, 2426, 6772, 24979 respectively.

See also A219023 for similar conjectures.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588, 2012.

Example

a(11)=2 since the 5 and 7 are the only primes p<11 with 66-p and 66+p both prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[6n-Prime[k]]==True&&PrimeQ[6n+Prime[k]]==True, 1, 0], {k, 1, PrimePi[n-1]}]
Do[Print[n, " ", a[n]], {n, 1, 20000}]

Crossrefs

Cf. A000040, A002375, A218754, A218585, A218654, A218656, A218825.

Sequence in context: A300647 A114811 A275675 * A270746 A043531 A297772

Adjacent sequences: A219022 A219023 A219024 * A219026 A219027 A219028

Keyword

nonn

Author

Zhi-Wei Sun, Nov 10 2012

Status

approved