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The number of "Fermi-Dirac primes" (A050376) that are noninfinitary divisors of n.
2

%I #6 Apr 28 2024 11:35:38

%S 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,

%T 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,

%U 0,0,0,1,0,0,1,1,0,0,0,2,2,0,0,1,0,0,0

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

%H Amiram Eldar, <a href="/A372332/b372332.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F 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)).

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

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

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

%K nonn,easy

%O 1,16

%A _Amiram Eldar_, Apr 28 2024