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%I #9 May 31 2020 21:57:11
%S 1,6,140,270,672,8190,30240,332640,14303520,17428320,27027000,
%T 191711520,2144862720,3506025600,5943057120,14378364000,45578332800,
%U 288662774400,505159855200,2020639420800,10680522652800,54557264361600
%N Harmonic numbers (A001599) k with a record abundancy index sigma(k)/k.
%C The corresponding record values are 1, 2, 2.4, 2.666..., 3, 3.2, 4, 4.363..., ...
%C The terms 1, 6, 672 and 30240 are multiply perfect numbers (A007691) with abundancy indices 1, 2, 3, and 4, respectively. There is no 5-multiperfect number (A046060) in this sequence since A046060(1) = 14182439040 is larger than the harmonic number 5943057120 which is 5-abundant, having an abundancy index 5.067...
%e The first 7 harmonic numbers are 1, 6, 28, 140, 270, 496 and 672. Their abundancy indices are 1, 2, 2, 2.4, 2.666..., 2 and 3. The record values, 1, 2, 2.4, 2.666... and 3 occur at 1, 6, 140, 270 and 672, the first 5 terms of this sequence.
%t rm = 0; s = {}; Do[h = DivisorSigma[0, n]/(r = DivisorSigma[1, n]/n); If[IntegerQ[h] && r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s
%Y Cf. A000203, A001599, A335316, A335317.
%K nonn,more
%O 1,2
%A _Amiram Eldar_, May 31 2020
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