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A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

%I A361316 #10 Mar 10 2023 10:37:50
%S A361316 1,4,3,8,5,2,7,32,9,20,11,12,13,7,5,32,17,12,19,8,21,22,23,16,25,52,
%T A361316 27,14,29,10,31,128,11,68,35,72,37,38,39,32,41,7,43,44,3,23,47,48,49,
%U A361316 100,17,104,53,18,55,56,57,116,59,4,61,31,63,256,65,11,67,136
%N A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.
%H A361316 Amiram Eldar, <a href="/A361316/b361316.txt">Table of n, a(n) for n = 1..10000</a>
%H A361316 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, No. 1 (1990), pp. 151-158.
%F A361316 a(n) = numerator(n*A037445(n)/A049417(n)).
%F A361316 a(n)/A361317(n) <= A099377(n)/A099378(n), with equality if and only if n is in A036537.
%F A361316 a(n)/A361317(n) >= A103339(n)/A103340(n), with equality if and only if n is in A138302.
%e A361316 Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...
%t A361316 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}]]; a[1] = 1; a[n_] := Numerator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]
%o A361316 (PARI) a(n) = {my(f = factor(n), b); numerator(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)))); }
%Y A361316 Cf. A036537, A037445, A049417, A077609, A063947, A138302, A361317 (denominators).
%Y A361316 Similar sequences: A099377, A103339.
%K A361316 nonn,frac
%O A361316 1,2
%A A361316 _Amiram Eldar_, Mar 09 2023