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A219157 Number of prime pairs {p,q} with p>q and p-2,q+2 also prime such that p+(1+mod(-n,6))q=n if n is not congruent to 2 mod 6, and p-q=n and q<n/2 if n=2 (mod 6). 10
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 0, 2, 0, 2, 2, 1, 1, 2, 3, 1, 0, 2, 1, 1, 0, 2, 2, 1, 2, 1, 2, 1, 0, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 0, 1, 3, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,22

COMMENTS

Conjecture: a(n)>0 for all n>30000 with n different from 38451, 46441, 50671, 62371.

This conjecture is stronger than the twin prime conjecture. It is similar to the conjecture associated with A219055 about sexy prime pairs.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

EXAMPLE

a(16)=1 since 16=7+3*3 with 7-2 and 3+2 prime. a(26)=1 since 26=31-5 with 31-2 and 5+2 prime.

MATHEMATICA

c[n_]:=c[n]=If[Mod[n-2, 6]==0, 1, -1-Mod[-n, 6]]

d[n_]:=d[n]=2+If[Mod[n-2, 6]>0, Mod[-n, 6], 0]

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+2]==True&&PrimeQ[n+c[n]Prime[k]]==True&&PrimeQ[n+c[n]Prime[k]-2]==True, 1, 0],

{k, 1, PrimePi[(n-1)/d[n]]}]

Do[Print[n, " ", a[n]], {n, 1, 100000}]

CROSSREFS

Cf. A001359, A006512, A002375, A046927, A219055, A218754, A218825, A219052.

Sequence in context: A194636 A286299 A081729 * A080215 A266871 A331290

Adjacent sequences:  A219154 A219155 A219156 * A219158 A219159 A219160

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Nov 12 2012

STATUS

approved

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Last modified February 18 15:30 EST 2020. Contains 332019 sequences. (Running on oeis4.)