A366073 The number of composite "Fermi-Dirac primes" (A082522) dividing n.

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

Offset: 1

Author

Amiram Eldar, Sep 28 2023

Keywords

• nonn
• easy

Comments

First differs from A071325 at n = 36.

The number of "Fermi-Dirac primes" that are infinitary divisors of n is A064547(n).

Links

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

Crossrefs

Cf. A000523, A050376, A064547, A071325, A082522, A366074.

Programs

Mathematica

f[p_, e_] := Floor[Log2[e]]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecsum(apply(exponent, factor(n)[, 2]));

Python

from sympy import factorint
def A366073(n): return -len(f:=factorint(n).values())+sum(map(int.bit_length,f)) # Chai Wah Wu, Feb 19 2025

Formula

Additive with a(p^e) = floor(log_2(e)) = A000523(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} P(2^k) = 0.53331724743088069672..., where P(s) is the prime zeta function.