cp's OEIS Frontend

A236456 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(n-k)/4 - 1, q = p + 2 and r = prime(q) + 2 are all prime, where phi(.) is Euler's totient function.

%I A236456 #6 Jan 26 2014 11:53:23
%S A236456 0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,2,3,3,2,4,2,2,4,5,4,2,3,1,3,2,3,
%T A236456 4,3,4,5,0,2,2,3,2,4,2,4,3,2,2,1,2,5,2,3,1,4,2,2,4,1,4,1,5,4,2,2,1,2,
%U A236456 1,5,3,3,1,2,2,4,1,3,4,2,2,1,0,2,4,2,1,3,1,4,3,5,3,2,1,3,2,3,2,0
%N A236456 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(n-k)/4 - 1, q = p + 2 and r = prime(q) + 2 are all prime, where phi(.) is Euler's totient function.
%C A236456 Conjecture: a(n) > 0 for all n > 357.
%C A236456 This is much stronger than the twin prime conjecture. Actually it implies that there are infinitely many primes p such that {p, p + 2} and {prime(p+2), prime(p+2) + 2} are both twin prime pairs. See A236457 for such primes p.
%H A236456 Zhi-Wei Sun, <a href="/A236456/b236456.txt">Table of n, a(n) for n = 1..10000</a>
%e A236456 a(18) = 1 since 18 = 3 + 15 with phi(3) + phi(15)/4 - 1  = 3,  3 + 2 = 5 and prime(5) + 2 = 13 all prime.
%e A236456 a(50) = 1 since 50 = 16 + 34 with phi(16) + phi(34)/4 - 1 = 11, 11 + 2 = 13 and prime(13) + 2 = 43 all prime.
%e A236456 a(929) = 1 since 929 = 441 + 488 with phi(441) + phi(488)/4 - 1 = 252 + 60 - 1 = 311, 311 + 2 = 313 and prime(313) + 2 = 2083 all prime.
%t A236456   p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n+2]+2]
%t A236456 f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/4-1
%t A236456 a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
%t A236456 Table[a[n],{n,1,100}]
%Y A236456 Cf. A000010, A000040, A001359, A006512, A236097, A236457, A236458.
%K A236456 nonn
%O A236456 1,19
%A A236456 _Zhi-Wei Sun_, Jan 26 2014