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A361782 Numerators of the harmonic means of the bi-unitary divisors of the positive integers.

%I A361782 #10 Mar 24 2023 09:33:29
%S A361782 1,4,3,8,5,2,7,32,9,20,11,12,13,7,5,64,17,12,19,8,21,22,23,16,25,52,
%T A361782 27,14,29,10,31,64,11,68,35,72,37,38,39,32,41,7,43,44,3,23,47,32,49,
%U A361782 100,17,104,53,18,55,56,57,116,59,4,61,31,63,384,65,11,67,136
%N A361782 Numerators of the harmonic means of the bi-unitary divisors of the positive integers.
%H A361782 Amiram Eldar, <a href="/A361782/b361782.txt">Table of n, a(n) for n = 1..10000</a>
%H A361782 Jozsef Sandor, <a href="https://arxiv.org/abs/1105.0294">On bi-unitary harmonic numbers</a>, arXiv:1105.0294 [math.NT], 2011.
%F A361782 a(n) = numerator(n*A286324(n)/A188999(n)).
%F A361782 f(n) = a(n)/A361783(n) is multiplicative with f(p^e) = (e+1)*(p-1)/(p^(e+1)-1) if e is odd, and e/((p^(e+1)-1)/(p-1) - p^(e/2)) if e is even.
%e A361782 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 A361782 f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]
%o A361782 (PARI) a(n) = {my(f = factor(n), p, e); numerator(n * prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  if(e%2, (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))))); }
%Y A361782 Cf. A188999, A222266, A286324, A361783 (denominators).
%Y A361782 Similar sequences: A099377, A103339, A361316.
%K A361782 nonn,frac
%O A361782 1,2
%A A361782 _Amiram Eldar_, Mar 24 2023