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.

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, 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, 10, 9, 6, 10, 7, 8, 12, 11, 10, 8, 11, 9, 12, 7, 13, 12, 13

Offset

1,22

Comments

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.

See also A236263 for a similar sequence.

Links

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

Example

a(23) = 1 since phi(7)/2 + phi(16)/8 = 3 + 1 = 4 with 4! - prime(4) = 24 - 7 = 17 prime.
a(26) = 1 since phi(9)/2 + phi(17)/8 = 3 + 2 = 5 with 5! - prime(5) = 120 - 11 = 109 prime.

Mathematica

q[n_]:=IntegerQ[n]&&PrimeQ[n!-Prime[n]]
f[n_, k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
a[n_]:=Sum[If[q[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.

Sequence in context: A280521 A278043 A014643 * A238645 A118382 A007723

Adjacent sequences: A236262 A236263 A236264 * A236266 A236267 A236268

Keyword

nonn

Author

Zhi-Wei Sun, Jan 21 2014

Status

approved