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

1,16

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

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A322338 A158971 A121467 * A362412 A071325 A064727

Adjacent sequences: A366070 A366071 A366072 * A366074 A366075 A366076

Keyword

nonn,easy

Author

Amiram Eldar, Sep 28 2023

Status

approved