cp's OEIS Frontend

A236074 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/6 + 1, prime(2*p) - 2*prime(p) and prime(p) - 2*prime((p-1)/2) are all prime}|, where phi(.) is Euler's totient function.

%I A236074 #10 Jan 19 2014 04:44:00
%S A236074 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,1,0,0,1,1,0,1,1,2,1,1,2,0,1,
%T A236074 0,0,0,0,2,2,0,1,0,0,1,1,0,1,0,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,
%U A236074 0,1,0,0,0,2,0,0,0,1,0,4,0,0,1,2,0,1,2,1,0,3,4,0,0,0,2,1,3,1,2,0
%N A236074 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/6 + 1, prime(2*p) - 2*prime(p) and prime(p) - 2*prime((p-1)/2) are all prime}|, where phi(.) is Euler's totient function.
%C A236074 Conjecture: (i) a(n) > 0 for all n > 116.
%C A236074 (ii) For any integer n > 196, there is a positive integer k < n such that p = phi(k) + phi(n-k)/6 + 1, prime(2*p) - 2*prime(p) and prime(p-1) - 2*prime((p-1)/2) are all prime.
%C A236074 Clearly, part (i) (or part (ii))  implies that there are infinitely many odd primes p with prime(2*p) - 2*prime(p) and prime(p) - 2*prime((p-1)/2) (or prime(p-1) - 2*prime((p-1)/2), resp.) both prime.
%H A236074 Zhi-Wei Sun, <a href="/A236074/b236074.txt">Table of n, a(n) for n = 1..10000</a>
%e A236074 a(30) = 1 since phi(4) + phi(26)/6 + 1 = 5, prime(2*5) - 2*prime(5) = 29 - 2*11 = 7 and prime(5) - 2*prime((5-1)/2) = 11 - 2*3 = 5 are all prime.
%e A236074 a(204) = 1 since phi(159) + phi(45)/6 + 1 = 109, prime(2*109) - 2*prime(109) = 1361 - 2*599 = 163, and prime(109) - 2*prime((109-1)/2) = 599 - 2*251 = 97 are all prime.
%t A236074 PQ[n_]:=PQ[n]=n>0&&PrimeQ[n]
%t A236074 p[n_]:=PQ[n]&&PQ[Prime[2n]-2Prime[n]]&&PQ[Prime[n]-2*Prime[(n-1)/2]]
%t A236074 f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/6+1
%t A236074 a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
%t A236074 Table[a[n],{n,1,100}]
%Y A236074 Cf. A000010, A000040, A234694, A235924, A236075.
%K A236074 nonn
%O A236074 1,29
%A A236074 _Zhi-Wei Sun_, Jan 19 2014