A366247 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366243.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Author

Amiram Eldar, Oct 05 2023

Keywords

• nonn
• easy

Comments

First differs from A101436 at n = 32.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A050376, A064547, A101436, A139352, A366242, A366243, A366246.

Programs

Mathematica

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

s(e) = if(e>3, s(e\4)) + e%4\2 \\ after Charles R Greathouse IV at A139352
a(n) = vecsum(apply(s, factor(n)[, 2]));

Formula

Additive with a(p^e) = A139352(e).

a(n) = A064547(n) - A366246(n).

a(n) = A064547(A366245(n)).

a(n) >= 0, with equality if and only if n is in A366242.

a(n) <= A064547(n), with equality if and only if n is in A366243.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.39310573826635831710..., where f(x) = Sum_{k>=0} (x^(2*4^k)/(1+x^(2*4^k))).