cp's OEIS Frontend

A235358 a(n) = |{0 < k < n: g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime with g(n,k) = phi(k) + phi(n-k)/8}|, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

%I A235358 #19 Jan 09 2014 23:09:42
%S A235358 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,1,0,0,0,0,0,
%T A235358 0,0,1,0,1,1,0,3,1,2,1,1,0,1,0,1,0,1,1,1,1,2,2,1,0,1,0,0,0,0,1,0,1,2,
%U A235358 0,2,1,2,0,1,0,1,1,0,0,0,0,0,2,2,0,1,0,0,1,1,0,0,0,0,0,0,0,0,1,1
%N A235358 a(n) = |{0 < k < n: g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime with g(n,k) = phi(k) + phi(n-k)/8}|, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).
%C A235358 Conjecture: a(n) > 0 for all n > 1234.
%C A235358 See also part (ii) of the conjecture in A235343.
%C A235358 We have verified the conjecture for n up to 100000.
%H A235358 Zhi-Wei Sun, <a href="/A235358/b235358.txt">Table of n, a(n) for n = 1..10000</a>
%e A235358 a(50) = 1 since phi(10) + phi(40)/4 = 6 with 6 - 1, 6 + 1 and q(6) - 1 = 3 all prime.
%t A235358 f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/8
%t A235358 p[n_,k_]:=PrimeQ[f[n,k]-1]&&PrimeQ[f[n,k]+1]&&PrimeQ[PartitionsQ[f[n,k]]-1]
%t A235358 a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
%t A235358 Table[a[n],{n,1,100}]
%Y A235358 Cf. A000009, A000010, A000040, A001359, A006512, A014574, A234514, A234567, A234615, A235343, A235344, A235346, A235356, A235357.
%K A235358 nonn
%O A235358 1,42
%A A235358 _Zhi-Wei Sun_, Jan 07 2014