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A361319 Indices of records in the sequence of infinitary harmonic means A361316(k)/A361317(k).

%I A361319 #9 Mar 10 2023 10:53:44
%S A361319 1,2,3,4,5,6,8,10,12,15,20,24,30,40,54,56,60,84,105,120,168,210,264,
%T A361319 270,280,360,420,540,660,756,840,1080,1320,1512,1848,1890,2310,2520,
%U A361319 3080,3640,3780,4620,5460,5940,7020,7560,9240,10920,11880,14040,16632,19656
%N A361319 Indices of records in the sequence of infinitary harmonic means A361316(k)/A361317(k).
%H A361319 Amiram Eldar, <a href="/A361319/b361319.txt">Table of n, a(n) for n = 1..200</a>
%e A361319 The infinitary harmonic means of the first 6 positive integers are 1 < 4/3 < 3/2 < 8/5 < 5/3 < 2. The next record, A361316(8)/A361317(8) = 32/15, occurs at 8. Therefore, the first 7 terms of this sequence are 1, 2, 3, 4, 5, 6 and 8.
%t A361319 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}]]; ihmean[1] = 1; ihmean[n_] := n*Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{ih, ihmax = 0, s = {}}, Do[ih = ihmean[k]; If[ih > ihmax, ihmax = ih; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[20000]
%o A361319 (PARI) 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 A361319 lista(kmax) = {my(ih, ihmax=0); for(k = 1, kmax, ih = ihmean(k); if(ih > ihmax, ihmax = ih; print1(k, ", ")));}
%Y A361319 Cf. A361316, A361317.
%Y A361319 Similar sequences: A179971, A348654.
%Y A361319 Other sequences related to records of infinitary divisors: A037992, A327634.
%K A361319 nonn
%O A361319 1,2
%A A361319 _Amiram Eldar_, Mar 09 2023