%I
%S 1,6,140,672,30240,332640,2178540,17428320,163390500,191711520,
%T 326781000,1307124000,2701389600,8104168800,14378364000,40520844000,
%U 1175104476000,7131668544000,49921679808000
%N Harmonic numbers (A001599) with a record number of divisors that are harmonic numbers.
%C The corresponding record values are 1, 2, 3, 4, 7, 9, 12, 15, 17, 19, 21, 25, 29, 34, 35, 43, 54, 75, 84.
%e 1 has one divisor which is a harmonic number: 1. 6 has 2 divisors which are harmonic numbers: 1 and 6. The next harmonic number with more than 2 divisors which are harmonic numbers is 140 with 3 such divisors: 1, 28 and 140.
%t h = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {_, _}][[;; , 2]]; hQ[n_] := Divisible[n * DivisorSigma[0, n], DivisorSigma[1, n]]; hsigma[n_] := DivisorSum[n, Boole @ hQ[#] &]; hm = 0; s = {}; Do[h1 = hsigma[h[[k]]]; If[h1 > hm, hm = h1; AppendTo[s, h[[k]]]], {k, 1, Length[h]}]; s
%Y Cf. A001599, A335317.
%K nonn,more
%O 1,2
%A _Amiram Eldar_, Jun 04 2020
