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%I #19 Apr 16 2018 03:02:57
%S 0,0,0,0,0,0,0,1,1,1,1,0,0,0,1,2,1,1,2,3,2,1,1,2,2,1,2,3,4,3,4,4,5,2,
%T 4,3,4,5,5,6,5,6,8,7,9,8,6,6,5,8,9,4,8,7,7,5,5,7,7,8,8,6,7,8,7,10,5,8,
%U 9,8,7,7,6,7,8,12,10,6,8,9,9,12,9,8,7,13
%N a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.
%C It might seem that a(n) > 0 for all n > 14, but a(7365) = 0. If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! + prime(m) prime.
%H Zhi-Wei Sun, <a href="/A236325/b236325.txt">Table of n, a(n) for n = 1..7000</a>
%e a(10) = 1 since phi(1)/2 + phi(9)/12 = 1/2 + 6/12 = 1 with 1! + prime(1) = 1 + 2 = 3 prime.
%e a(23) = 1 since phi(10)/2 + phi(13)/12 = 2 + 1 = 3 with 3! + prime(3) = 6 + 5 = 11 prime.
%t p[n_]:=IntegerQ[n]&&PrimeQ[n!+Prime[n]]
%t f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12
%t a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236263, A236265.
%K nonn
%O 1,16
%A _Zhi-Wei Sun_, Jan 22 2014
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