%I
%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 5multiperfect number (A046060) in this sequence since A046060(1) = 14182439040 is larger than the harmonic number 5943057120 which is 5abundant, 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
