This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A236265 #25 Jan 22 2014 06:08:32
%S A236265 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,2,2,1,2,2,4,3,5,1,3,2,
%T A236265 3,3,4,5,9,5,5,6,7,8,8,8,5,7,5,8,8,5,5,9,8,6,6,9,8,10,6,9,4,6,9,9,8,
%U A236265 10,9,6,10,7,8,12,11,10,8,11,9,12,7,13,12,13
%N A236265 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 A236265 It seems that a(n) > 0 for all n > 21. If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! - prime(m) prime.
%C A236265 See also A236263 for a similar sequence.
%H A236265 Zhi-Wei Sun, <a href="/A236265/b236265.txt">Table of n, a(n) for n = 1..7000</a>
%e A236265 a(23) = 1 since phi(7)/2 + phi(16)/8 = 3 + 1 = 4 with 4! - prime(4) = 24 - 7 = 17 prime.
%e A236265 a(26) = 1 since phi(9)/2 + phi(17)/8 = 3 + 2 = 5 with 5! - prime(5) = 120 - 11 = 109 prime.
%t A236265 q[n_]:=IntegerQ[n]&&PrimeQ[n!-Prime[n]]
%t A236265 f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
%t A236265 a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-1}]
%t A236265 Table[a[n],{n,1,100}]
%Y A236265 Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236263.
%K A236265 nonn
%O A236265 1,22
%A A236265 _Zhi-Wei Sun_, Jan 21 2014