

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
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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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 20122015.


EXAMPLE

a(19) = 1 since 13, 19(131) = 7 and 19+(131) = 31 are all prime.
a(142) = 1 since 41, 142(2*411) = 61 and 142+(2*411) = 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

ZhiWei Sun, Aug 27 2015


STATUS

approved



