A372332 The number of "Fermi-Dirac primes" (A050376) that are noninfinitary divisors of n.

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

Offset: 1

Author

Amiram Eldar, Apr 28 2024

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A023416, A050376, A064547, A348341, A372331.

Programs

Mathematica

f[p_, e_] := DigitCount[e, 2, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecsum(apply(x -> logint(x, 2) + 1 - hammingweight(x), factor(n)[, 2]));

Formula

Additive with a(p^e) = A023416(e).

a(n) = log_2(A372331(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.39726277693465233149..., where f(x) = Sum_{k>=0} x^(2^(k+1))/(1+x^(2^k)).