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A361385 a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.

%I A361385 #10 Mar 10 2023 10:53:40
%S A361385 0,2,2,3,3,4,4,4,5,5,5,4,3,5,5,5,4,6,5,5,6,6,5,6,5,6,6,6,5,7,4,5,5,6,
%T A361385 7,6,6,6,7,6,6,7,6,6,6,7,6,8,7,7,7,6,7,7,7,6,8,6,5,6,7,6,7,7,6,8,7,7,
%U A361385 8,7,6,7,8,7,6,8,7,7,7,7,9,6,8,6,8,8,7
%N A361385 a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.
%C A361385 Each term appears a finite number of times in the sequence (Hagis and Cohen, 1990).
%H A361385 Amiram Eldar, <a href="/A361385/b361385.txt">Table of n, a(n) for n = 1..239</a>
%H A361385 Peter Hagis, Jr. and Graeme L. Cohen, <a href="http://dx.doi.org/10.1017/S0004972700017949">Infinitary harmonic numbers</a>, Bull. Australian math. Soc., Vol. 41 (1990), pp. 151-158.
%F A361385 a(n) = A064547(A063947(n)).
%t A361385 f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; ih[1] = 1; ih[n_] := n*Times @@ f @@@ FactorInteger[n]; ic[n_] := Plus @@ (DigitCount[Last /@ FactorInteger[n], 2, 1]); ic[1] = 0; ic /@ Select[Range[10^5], IntegerQ[ih[#]] &]
%o A361385 (PARI) A064547(n) = {my(f = factor(n)[, 2]); sum(k=1, #f, hammingweight(f[k])); } \\ _Michel Marcus_ at A064547
%o A361385 ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
%o A361385 lista(kmax) = {my(ih); for(k = 1, kmax, ih = ihmean(k); if(denominator(ih) == 1, print1(A064547(k), ", ")));}
%Y A361385 Cf. A006086, A006087, A361384 (analogous unitary sequence).
%K A361385 nonn
%O A361385 1,2
%A A361385 _Amiram Eldar_, Mar 10 2023