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A179971 Positions of records in the sequence of harmonic means, i.e., in the sequence of rationals A099377(.)/A099378(.).

%I A179971 #14 Feb 16 2025 08:33:12
%S A179971 1,2,3,4,6,8,10,12,16,18,20,24,30,36,48,60,72,84,90,108,120,144,168,
%T A179971 180,240,336,360,420,480,504,630,720,840,1008,1080,1260,1440,1680,
%U A179971 2160,2520,3360,3780,3960,4200,4320,4620,5040,7560,9240,10080,12600,13860,15120
%N A179971 Positions of records in the sequence of harmonic means, i.e., in the sequence of rationals A099377(.)/A099378(.).
%H A179971 Amiram Eldar, <a href="/A179971/b179971.txt">Table of n, a(n) for n = 1..200</a>
%H A179971 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OresConjecture.html">Ore's Conjecture</a>
%e A179971 The sequence of harmonic means starts 1 < 4/3 < 3/2 < 12/7, increasing from the first to the fourth, which adds 1 to 4 to the sequence.
%e A179971 The fifth harmonic mean is 5/3, smaller than 12/7 and not a record, so 5 is not in the sequence.
%p A179971 hm := proc(n) option remember; n* numtheory[tau](n)/numtheory[sigma](n) ; end proc:
%p A179971 A179971 := proc(n) option remember; if n = 1 then 1; else for k from procname(n-1)+1 do if hm(k) > hm(procname(n-1)) then return k; end if; end do; end if; end proc:
%p A179971 seq(A179971(n),n=1..40) ; # _R. J. Mathar_, Aug 06 2010
%t A179971 f[n_] := f[n] = DivisorSigma[0, n]/Plus @@ (1/Divisors@n); k = 1; mx = 0; lst = {}; While[k < 18480, a = f@k; If[a > mx, mx = a; AppendTo[lst, k]]; k++ ]; lst
%Y A179971 Cf. A099377, A099378.
%K A179971 nonn
%O A179971 1,2
%A A179971 _Robert G. Wilson v_, Aug 04 2010
%E A179971 Definition rephrased by _R. J. Mathar_, Aug 06 2010