A236344 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2^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, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 3, 4, 2, 4, 3, 5, 2, 3, 4, 3, 3, 4, 6, 5, 6, 6, 7, 7, 5, 4, 6, 6, 5, 7, 5, 3, 3, 3, 7, 4, 5, 5, 8, 4, 6, 5, 5, 5, 6, 4, 5, 4, 5, 4, 3, 4, 5, 6, 3, 6, 9, 6, 9, 8, 13, 5, 11, 5, 6, 7, 11, 4, 9, 9, 5, 6, 6, 11, 7, 8, 9, 9, 4

Offset

1,20

Comments

a(n) = 0 for n = 1, ..., 15, 481, 564, 66641, 70965, 72631, .... If a(n) > 0 infinitely often, then there are infinitely many positive integers m with 2^m + prime(m) prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(26) = 1 since phi(5)/2 + phi(21)/12 = 2 + 1 = 3 with 2^3 + prime(3) = 8 + 5 = 13 prime.
a(5907) = 1 since phi(3944)/2 + phi(5907-3944)/12 = 896 + 150 = 1046 with 2^(1046) + prime(1046) = 2^(1046) + 8353 prime.

Mathematica

p[n_]:=IntegerQ[n]&&PrimeQ[2^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, A000079, A077375, A236241, A236256.

Sequence in context: A100727 A366134 A064892 * A333124 A274109 A083019

Adjacent sequences: A236341 A236342 A236343 * A236345 A236346 A236347

Keyword

nonn

Author

Zhi-Wei Sun, Jan 22 2014

Status

approved