cp's OEIS Frontend

A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function.

%I A236097 #13 Apr 06 2014 04:11:49
%S A236097 0,0,0,0,0,0,0,2,2,1,3,1,1,2,2,3,0,1,1,1,0,0,0,1,0,0,0,1,1,2,0,5,5,2,
%T A236097 4,1,5,3,3,2,4,4,9,5,9,4,10,3,6,6,8,5,10,4,4,7,8,10,5,8,9,9,4,11,3,5,
%U A236097 5,9,5,4,4,5,6,8,7,6,3,11,4,8,10,9,8,7,6,11,7,9,4,6,5,6,2,9,4,7,6,7,10,9
%N A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function.
%C A236097 Conjecture: a(n) > 0 for all n > 31.
%C A236097 This implies that there are infinitely many primes p with {prime(p) - p - 1, prime(p) - p + 1} a twin prime pair.
%H A236097 Zhi-Wei Sun, <a href="/A236097/b236097.txt">Table of n, a(n) for n = 1..10000</a>
%H A236097 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A236097 a(20) = 1 since phi(2) + phi(18)/2 + 1 = 5, prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are all prime.
%e A236097 a(36) = 1 since phi(21) + phi(15)/2 + 1 = 17, prime(17) - 17 - 1 = 41 and prime(17) - 17 + 1 = 43 are all prime.
%t A236097 p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
%t A236097 f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1
%t A236097 a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
%t A236097 Table[a[n],{n,1,100}]
%Y A236097 Cf. A000010, A000040, A001359, A006512, A014574, A234694, A234695, A235924, A236074, A236119.
%K A236097 nonn
%O A236097 1,8
%A A236097 _Zhi-Wei Sun_, Jan 19 2014