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A219025 Number of primes p<n such that 6n-p 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)