A218867 Number of prime pairs {p,q} with p>q and {p-4,q+4} also prime such that p+(1+(n mod 6))q=n if n is not congruent to 4 (mod 6), and p-q=n and q<n/2 if n=4 (mod 6).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 1, 2, 2, 2, 2, 0, 2, 2, 1, 1, 1, 2, 1, 0, 0, 1, 0, 2, 2, 0, 2, 1, 3, 0, 1, 1, 2, 2, 1, 0, 3, 2, 3, 0, 2, 1, 4, 1, 1, 2, 1, 3, 2

Offset

1,30

Comments

Conjecture: a(n)>0 for all n>50000 with n different from 50627, 61127, 66503.

This conjecture implies that there are infinitely many cousin prime pairs. It is similar to the conjectures related to A219157 and A219055.

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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10^5.

Example

a(20)=1 since 20=11+3*3 with 11-4 and 3+4 prime. a(28)=1 since 28=41-13 with 41-4 and 13+4 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]+4] == True && PrimeQ[n+c[n]Prime[k]] == True && PrimeQ[n+c[n]Prime[k]-4]==True, 1, 0], {k, 1, PrimePi[(n-1)/d[n]]}]; Do[Print[n, " ", a[n]], {n, 100}]

Crossrefs

Cf. A023200, A046132, A219157, A219055, A002375, A046927, A218754, A218825, A219052.

Sequence in context: A262115 A071460 A369862 * A295664 A250213 A033794

Adjacent sequences: A218864 A218865 A218866 * A218868 A218869 A218870

Keyword

nonn

Author

Zhi-Wei Sun, Nov 13 2012

Status

approved