cp's OEIS Frontend

A236417 a(n) = |{0 < k < n: p = phi(k)/2 + phi(n-k)/12 + 1 and A047967(p) are both prime}|.

%I A236417 #7 Apr 06 2014 10:47:09
%S A236417 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A236417 0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0,0,0,2,0,0,2,1,0,1,0,3,1,0,1,1,
%U A236417 1,2,1,2,0,1,2,2,2,1,2,1,1,3,1,1,4,2,0,1,3,2,2,0,2,2,4,2,3,0,3,2
%N A236417 a(n) = |{0 < k < n: p = phi(k)/2 + phi(n-k)/12 + 1 and A047967(p) are both prime}|.
%C A236417 Conjecture: a(n) > 0 for all n > 98.
%C A236417 We have verified this for n up to 36000.
%C A236417 The conjecture implies that there are infinitely many primes p with A047967(p) prime.
%H A236417 Zhi-Wei Sun, <a href="/A236417/b236417.txt">Table of n, a(n) for n = 1..10000</a>
%H A236417 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A236417 a(36) = 1 since phi(23)/2 + phi(13)/12 + 1 = 13 with A047967(13) = 83 prime.
%e A236417 a(71) = 1 since phi(43)/2 + phi(28)/12 + 1 = 23 with A047967(23) = 1151 prime.
%t A236417 pq[n_]:=PrimeQ[n]&&PrimeQ[PartitionsP[n]-PartitionsQ[n]]
%t A236417 f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12+1
%t A236417 a[n_]:=Sum[If[pq[f[n,k]],1,0],{k,1,n-1}]
%t A236417 Table[a[n],{n,1,100}]
%Y A236417 Cf. A000010, A000040, A047967.
%K A236417 nonn
%O A236417 1,56
%A A236417 _Zhi-Wei Sun_, Jan 25 2014