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A361785 Indices of records in the sequence of bi-unitary harmonic means A361782(k)/A361783(k).

%I A361785 #10 Mar 24 2023 11:14:36
%S A361785 1,2,3,4,5,6,8,10,12,15,20,24,30,40,54,56,60,84,96,120,168,210,240,
%T A361785 270,280,360,420,480,672,840,1080,1320,1512,1680,1890,2160,2310,2520,
%U A361785 3080,3360,4320,5280,6048,7392,7560,9240,10920,11880,14040,15120,18480,20790
%N A361785 Indices of records in the sequence of bi-unitary harmonic means A361782(k)/A361783(k).
%H A361785 Amiram Eldar, <a href="/A361785/b361785.txt">Table of n, a(n) for n = 1..192</a>
%e A361785 The harmonic means of the bi-unitary divisors of the first 6 positive integers are 1 < 4/3 < 3/2 < 8/5 < 5/3 < 2. A361782(7)/A361783(7) = 9/5 < 2, and the next record, A361782(8)/A361783(8) = 32/15, occurs at 8. Therefore, the first 7 terms of this sequence are 1, 2, 3, 4, 5, 6 and 8.
%t A361785 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))]; buhmean[1] = 1; buhmean[n_] := Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{buh, buhmax = 0, s = {}}, Do[buh = buhmean[k]; If[buh > buhmax, buhmax = buh; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[20000]
%o A361785 (PARI) buhmean(n) = {my(f = factor(n), p, e); 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)))); }
%o A361785 lista(kmax) = {my(buh, buhmax=0); for(k = 1, kmax, buh = buhmean(k); if(buh > buhmax, buhmax = buh; print1(k, ", "))); }
%Y A361785 Cf. A361782, A361783.
%Y A361785 Similar sequences: A179971, A348654, A361319.
%Y A361785 Other sequences related to records of bi-unitary divisors: A293185, A292983, A292984.
%K A361785 nonn
%O A361785 1,2
%A A361785 _Amiram Eldar_, Mar 24 2023