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The number of "Fermi-Dirac primes" (A050376) that divide n.
1

%I #8 May 04 2024 05:34:07

%S 0,1,1,2,1,2,1,2,2,2,1,3,1,2,2,3,1,3,1,3,2,2,1,3,2,2,2,3,1,3,1,3,2,2,

%T 2,4,1,2,2,3,1,3,1,3,3,2,1,4,2,3,2,3,1,3,2,3,2,2,1,4,1,2,3,3,2,3,1,3,

%U 2,3,1,4,1,2,3,3,2,3,1,4,3,2,1,4,2,2,2

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

%C Differs from A345222 at n = 64, 128, 192, 320, 384, ... .

%H Amiram Eldar, <a href="/A372502/b372502.txt">Table of n, a(n) for n = 1..10000</a>

%F Additive with a(p^e) = A070939(e).

%F a(n) = A064547(n) + A372332(n).

%F Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=1} P(2^k) = 0.53331724743088069672..., and P(s) is the prime zeta function.

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

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

%Y Cf. A050376, A064547, A070939, A077761, A345222, A372332.

%K nonn,easy

%O 1,4

%A _Amiram Eldar_, May 04 2024