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%I #23 Jan 22 2014 03:59:15
%S 0,0,0,0,0,1,1,0,1,1,1,1,1,2,1,0,0,1,2,3,3,4,5,4,4,5,7,4,5,6,6,5,5,5,
%T 7,6,7,9,7,8,7,7,5,11,8,8,8,11,8,7,5,10,6,9,8,10,7,8,10,9,7,8,9,13,8,
%U 8,9,10,6,11,10,7,7,9,11,13,8,11,13,11,14,6
%N a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.
%C It seems that a(n) > 0 for all n > 17. (We have verified this for n up to 13000.) If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! + prime(m) prime.
%C See also A236265 for a similar sequence.
%H Zhi-Wei Sun, <a href="/A236263/b236263.txt">Table of n, a(n) for n = 1..8100</a>
%e a(18) = 1 since phi(3)/2 + phi(15)/8 = 1 + 1 = 2 with 2! + prime(2) = 2 + 3 = 5 prime.
%e a(356) = 1 since phi(203)/2 + phi(153)/8 = 84 + 12 = 96 with 96! + prime(96) = 96! + 503 prime.
%e a(457) = 1 since phi(7)/2 + phi(450)/8 = 3 + 15 = 18 with 18! + prime(18) = 18! + 61 = 6402373705728061 prime.
%t q[n_]:=IntegerQ[n]&&PrimeQ[n!+Prime[n]]
%t f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
%t a[n_]:=Sum[If[q[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, A236265.
%K nonn
%O 1,14
%A _Zhi-Wei Sun_, Jan 21 2014
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