A372502 The number of "Fermi-Dirac primes" (A050376) that divide n.

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 3, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2

Offset

1,4

Comments

Differs from A345222 at n = 64, 128, 192, 320, 384, ... .

Links

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

Formula

Additive with a(p^e) = A070939(e).

a(n) = A064547(n) + A372332(n).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=1} P(2^k) = 0.53331724743088069672..., and P(s) is the prime zeta function.

Mathematica

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

Prog

(PARI) a(n) = vecsum(apply(x -> exponent(x) + 1, factor(n)[, 2]));
(Python)
from sympy import factorint
def A372502(n): return sum(e.bit_length() for e in factorint(n).values()) # Chai Wah Wu, Feb 18 2025

Crossrefs

Cf. A050376, A064547, A070939, A077761, A345222, A372332.

Sequence in context: A317751 A106490 A349281 * A345222 A327399 A122375

Adjacent sequences: A372499 A372500 A372501 * A372503 A372504 A372505

Keyword

nonn,easy

Author

Amiram Eldar, May 04 2024

Status

approved