A236325 - OEIS

A236325

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.

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, 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, 9, 8, 7, 7, 6, 7, 8, 12, 10, 6, 8, 9, 9, 12, 9, 8, 7, 13

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OFFSET

1,16

COMMENTS

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.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

EXAMPLE

a(10) = 1 since phi(1)/2 + phi(9)/12 = 1/2 + 6/12 = 1 with 1! + prime(1) = 1 + 2 = 3 prime.

a(23) = 1 since phi(10)/2 + phi(13)/12 = 2 + 1 = 3 with 3! + prime(3) = 6 + 5 = 11 prime.

MATHEMATICA

p[n_]:=IntegerQ[n]&&PrimeQ[n!+Prime[n]]

f[n_, k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12

a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-1}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236263, A236265.