cp's OEIS Frontend

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

%I A199920 #22 Jul 22 2025 13:26:35
%S A199920 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,
%T A199920 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,
%U A199920 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
%N A199920 Number of ways to write n = p+k with p, p+6, 6k-1 and 6k+1 all prime.
%C A199920 Conjecture: a(n)>0 for all n>11.
%C A199920 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.
%C A199920 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.
%H A199920 Zhi-Wei Sun, <a href="/A199920/b199920.txt">Table of n, a(n) for n = 1..50000</a>
%H A199920 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A199920 a(21)=1 since 21=11+10 with 11, 11+6, 6*10-1 and 6*10+1 all prime.
%t A199920 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]}]
%t A199920 Do[Print[n," ",a[n]],{n,1,100}]
%t A199920 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 *)
%o A199920 (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
%Y A199920 Cf. A001359, A006512, A023201, A199800, A219055, A219864, A219923, A002375.
%K A199920 nonn,nice
%O A199920 1,8
%A A199920 _Zhi-Wei Sun_, Dec 22 2012