A199920 Number of ways to write n = p+k with p, p+6, 6k-1 and 6k+1 all prime

0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 0, 3, 1, 3, 2, 2, 2, 3, 2, 2, 1, 2, 3, 3, 3, 1, 1, 3, 2, 4, 1, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 3, 5, 3, 3, 3, 3, 4, 5, 3, 3, 3, 3, 5, 4, 4, 3, 4, 3, 3, 2, 3, 6, 5, 4, 2, 1, 3, 5, 5, 5, 2, 2, 3, 5, 3, 5, 4, 5, 2, 3, 2, 5, 5, 6, 4, 2, 3, 3, 4, 3, 3, 5, 4, 3, 1, 1, 4, 5, 7

Offset

1,8

Comments

Conjecture: a(n)>0 for all n>11.

This implies that there are infinitely many twin primes and also infinitely many sexy primes. It has been verified for n up to 10^9. See also A199800 for a weaker version of this conjecture.

Zhi-Wei Sun also conjectured that any integer n>6 not equal to 319 can be written as p+k with p, p+6, 3k-2+(n mod 2) and 3k+2-(n mod 2) all prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..50000

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

Example

a(21)=1 since 21=11+10 with 11, 11+6, 6*10-1 and 6*10+1 all prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[6(n-Prime[k])-1]==True&&PrimeQ[6(n-Prime[k])+1]==True, 1, 0], {k, 1, PrimePi[n]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
Table[Count[Table[{n-i, i}, {i, n-1}], _?(AllTrue[{#[[1]], #[[1]]+6, 6#[[2]]-1, 6#[[2]]+1}, PrimeQ]&)], {n, 100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 19 2015 *)

Prog

(PARI) a(n)=my(s, p=2, q=3); forprime(r=5, n+5, if(r-p==6 && isprime(6*n-6*p-1) && isprime(6*n-6*p+1), s++); if(r-q==6 && isprime(6*n-6*q-1) && isprime(6*n-6*q+1), s++); p=q; q=r); s \\ Charles R Greathouse IV, Jul 31 2016

Crossrefs

Cf. A001359, A006512, A023201, A199800, A219055, A219864, A219923, A002375.

Sequence in context: A274576 A257081 A271484 * A177995 A332104 A238735

Adjacent sequences: A199917 A199918 A199919 * A199921 A199922 A199923

Keyword

nonn,nice

Author

Zhi-Wei Sun, Dec 22 2012

Status

approved