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%I #8 Dec 25 2013 02:24:24
%S 0,0,0,0,0,0,0,1,1,1,2,1,3,2,3,4,2,6,3,6,6,5,7,4,6,4,5,7,9,4,6,4,10,7,
%T 2,11,9,12,6,9,10,9,12,11,10,6,12,13,8,11,9,10,7,8,7,11,8,9,6,14,4,15,
%U 5,14,7,15,5,12,11,9,10,9,10,8,10,7,12,11,15,10
%N a(n) = |{0 < k < n: 2^{phi(k)/2}*3^{phi(n-k)/4} + 1 is prime}|, where phi(.) is Euler's totient function.
%C Conjecture: a(n) > 0 for all n > 7.
%C This implies that there are infinitely many primes of the form 2^k*3^m + 1, where k and m are positive integers.
%H Zhi-Wei Sun, <a href="/A234361/b234361.txt">Table of n, a(n) for n = 1..5000</a>
%e a(10) = 1 since 2^{phi(5)/2}*3^{phi(5)/4} + 1 = 13 is prime.
%e a(12) = 1 since 2^{phi(4)/2}*3^{phi(8)/4} + 1 = 13 is prime.
%e a(35) = 2 since 2^{phi(3)/2}*3^{phi(32)/4} + 1 = 2*3^4 + 1 = 163 and 2^{phi(5)/2}*3^{phi(30)/4} + 1 = 2^2*3^2 + 1 = 37 are both prime.
%t f[n_,k_]:=f[n,k]=2^(EulerPhi[k]/2)*3^(EulerPhi[n-k]/4)+1
%t a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000010, A000040, A000079, A000244, A234309, A234310, A234337, A234344, A234346, A234347, A234359, A234360.
%K nonn
%O 1,11
%A _Zhi-Wei Sun_, Dec 24 2013
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