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A366246 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366242.

%I A366246 #9 Oct 06 2023 10:56:47
%S A366246 0,1,1,0,1,2,1,1,0,2,1,1,1,2,2,1,1,1,1,1,2,2,1,2,0,2,1,1,1,3,1,2,2,2,
%T A366246 2,0,1,2,2,2,1,3,1,1,1,2,1,2,0,1,2,1,1,2,2,2,2,2,1,2,1,2,1,1,2,3,1,1,
%U A366246 2,3,1,1,1,2,1,1,2,3,1,2,1,2,1,2,2,2,2
%N A366246 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366242.
%H A366246 Amiram Eldar, <a href="/A366246/b366246.txt">Table of n, a(n) for n = 1..10000</a>
%F A366246 Additive with a(p^e) = A139351(e).
%F A366246 a(n) = A064547(n) - A366247(n).
%F A366246 a(n) = A064547(A366244(n)).
%F A366246 a(n) >= 0, with equality if and only if n is in A366243.
%F A366246 a(n) <= A064547(n), with equality if and only if n is in A366242.
%F A366246 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = -0.25705126777012995187..., where f(x) = - x + Sum_{k>=0} (x^(4^k)/(1+x^(4^k))).
%t A366246 s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[OddQ[Mod[n, 4]], 1, 0]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366246 (PARI) s(e) = if(e>3, s(e\4)) + e%2 \\ after _Charles R Greathouse IV_ at A139351
%o A366246 a(n) = vecsum(apply(s, factor(n)[, 2]));
%Y A366246 Cf. A050376, A064547, A077761, A139351, A366242, A366243, A366247.
%K A366246 nonn,easy
%O A366246 1,6
%A A366246 _Amiram Eldar_, Oct 05 2023