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A372332
The number of "Fermi-Dirac primes" (A050376) that are noninfinitary divisors of n.
2
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
1,16
LINKS
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)).
MATHEMATICA
f[p_, e_] := DigitCount[e, 2, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecsum(apply(x -> logint(x, 2) + 1 - hammingweight(x), factor(n)[, 2]));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Apr 28 2024
STATUS
approved