A219025 Number of primes p

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

Author

Zhi-Wei Sun, Nov 10 2012

Keywords

• nonn

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

See also A219023 for similar conjectures.

Examples

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

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.

Crossrefs

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

Programs

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}]

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