cp's OEIS Frontend

A234504 Number of ways to write n = k + m with k > 0 and m > 0 such that 2^(phi(k) + phi(m)/4) - 5 is prime, where phi(.) is Euler's totient function.

%I A234504 #24 Jan 24 2014 01:14:20
%S A234504 0,0,0,0,0,0,0,1,1,0,2,1,1,3,2,3,2,3,2,3,4,5,5,4,5,6,7,4,5,6,7,6,5,7,
%T A234504 8,5,7,9,8,8,6,8,7,10,7,10,10,9,9,8,9,10,5,10,10,9,10,10,9,10,9,7,12,
%U A234504 14,10,9,5,11,7,13,8,13,6,9,11,11,14,15,9,13
%N A234504 Number of ways to write n = k + m with k > 0 and m > 0 such that 2^(phi(k) + phi(m)/4) - 5 is prime, where phi(.) is Euler's totient function.
%C A234504 Conjecture: a(n) > 0 for all n > 10.
%C A234504 We have verified this for n up to 50000. The conjecture implies that there are infinitely many primes of the form 2^n - 5.
%H A234504 Zhi-Wei Sun, <a href="/A234504/b234504.txt">Table of n, a(n) for n = 1..10000</a>
%e A234504 a(15) = 2 since 2^(phi(2) + phi(13)/4) - 5 = 2^4 - 5 = 11 and 2^(phi(3) + phi(12)/4) - 5 = 2^3 - 5 = 3 are both prime.
%t A234504 f[n_,k_]:=2^(EulerPhi[k]+EulerPhi[n-k]/4)-5
%t A234504 a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
%t A234504 Table[a[n],{n,1,100}]
%Y A234504 Cf. A000010, A000040, A000079, A050522, A156560, A234309, A234451, A234470, A234475, A234503, A236358.
%K A234504 nonn
%O A234504 1,11
%A A234504 _Zhi-Wei Sun_, Dec 26 2013