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The number of composite "Fermi-Dirac primes" (A082522) dividing n.
2

%I #12 Sep 29 2023 06:31:38

%S 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,

%T 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,

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

%N The number of composite "Fermi-Dirac primes" (A082522) dividing n.

%C First differs from A071325 at n = 36.

%C The number of "Fermi-Dirac primes" that are infinitary divisors of n is A064547(n).

%H Amiram Eldar, <a href="/A366073/b366073.txt">Table of n, a(n) for n = 1..10000</a>

%F Additive with a(p^e) = floor(log_2(e)) = A000523(e).

%F 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.

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

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

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

%K nonn,easy

%O 1,16

%A _Amiram Eldar_, Sep 28 2023