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A261627 Number of primes p such that n-(p*n'-1) and n+(p*n'-1) are both prime, where n' is 1 or 2 according as n is odd or even. 4
0, 0, 0, 0, 1, 0, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 1, 2, 2, 4, 2, 3, 2, 2, 1, 2, 2, 3, 1, 3, 2, 2, 3, 3, 3, 3, 3, 3, 1, 4, 1, 3, 2, 3, 4, 4, 3, 3, 2, 4, 3, 6, 2, 3, 2, 2, 3, 5, 3, 4, 4, 4, 2, 5, 4, 6, 1, 4, 2, 4, 3, 5, 4, 3, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Conjecture: a(n) > 0 for all n > 6, and a(n) = 1 only for n = 5, 7, 10, 11, 12, 19, 22, 30, 34, 44, 46, 72, 142.

This is stronger than Goldbach's conjecture (A002375) and Lemoine's conjecture (A046927).

I have verified the conjecture for n up to 10^8.

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 [math.NT], 2012-2015.

EXAMPLE

a(19) = 1 since 13, 19-(13-1) = 7 and 19+(13-1) = 31 are all prime.

a(142) = 1 since 41, 142-(2*41-1) = 61 and 142+(2*41-1) = 223 are all prime.

MATHEMATICA

Do[r=0; Do[If[PrimeQ[n-(3+(-1)^n)/2*Prime[k]+1]&&PrimeQ[n+(3+(-1)^n)/2*Prime[k]-1], r=r+1], {k, 1, PrimePi[2n/(3+(-1)^n)]}]; Print[n, " ", r]; Continue, {n, 1, 80}]

CROSSREFS

Cf. A000040, A002372, A002375, A046927, A219055, A237284, A261628.

Sequence in context: A201208 A006513 A105224 * A237112 A238013 A303940

Adjacent sequences:  A261624 A261625 A261626 * A261628 A261629 A261630

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 27 2015

STATUS

approved

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Last modified October 15 07:51 EDT 2019. Contains 328026 sequences. (Running on oeis4.)