

A236263


a(n) = {0 < k < n: m = phi(k)/2 + phi(nk)/8 is an integer with m! + prime(m) prime}, where phi(.) is Euler's totient function.


3



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, 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, 8, 9, 10, 6, 11, 10, 7, 7, 9, 11, 13, 8, 11, 13, 11, 14, 6
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OFFSET

1,14


COMMENTS

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.
See also A236265 for a similar sequence.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..8100


EXAMPLE

a(18) = 1 since phi(3)/2 + phi(15)/8 = 1 + 1 = 2 with 2! + prime(2) = 2 + 3 = 5 prime.
a(356) = 1 since phi(203)/2 + phi(153)/8 = 84 + 12 = 96 with 96! + prime(96) = 96! + 503 prime.
a(457) = 1 since phi(7)/2 + phi(450)/8 = 3 + 15 = 18 with 18! + prime(18) = 18! + 61 = 6402373705728061 prime.


MATHEMATICA

q[n_]:=IntegerQ[n]&&PrimeQ[n!+Prime[n]]
f[n_, k_]:=EulerPhi[k]/2+EulerPhi[nk]/8
a[n_]:=Sum[If[q[f[n, k]], 1, 0], {k, 1, n1}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236265.
Sequence in context: A301573 A061670 A236412 * A337126 A319572 A108063
Adjacent sequences: A236260 A236261 A236262 * A236264 A236265 A236266


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 21 2014


STATUS

approved



