

A219025


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


2



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
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OFFSET

1,11


COMMENTS

Conjecture: a(n)>0 for all n=6,7,...
This has been verified for n up to 10^8.
ZhiWei Sun also made the following general conjecture:
Let P(x) be any nonconstant integervalued 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

ZhiWei Sun, Table of n, a(n) for n = 1..20000
ZhiWei 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 66p and 66+p both prime.


MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[6nPrime[k]]==True&&PrimeQ[6n+Prime[k]]==True, 1, 0], {k, 1, PrimePi[n1]}]
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

ZhiWei Sun, Nov 10 2012


STATUS

approved



