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
KEYWORD
nonn,nice
AUTHOR
Zhi-Wei Sun, Dec 22 2012
STATUS
approved