This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A335370 #11 Jun 03 2020 23:56:52
%S A335370 1,28,1638,6200,2457000,4713984,1381161600,10200236032
%N A335370 Harmonic numbers m with a record number k of distinct prime numbers p_i (i = 1..k) that do not divide m such that m*p_1, m*p_1*p_2, ... , m*p_1*...*p_k are all harmonic numbers.
%C A335370 If m is a harmonic number (A001599), then it is possible to generate a new harmonic number m*p if p is a prime number that does not divide m and (p+1)/2 is a divisor of the harmonic mean of the divisors of m, h(m) = m * tau(m)/sigma(m) = m * A000005(m)/A000203(m).
%C A335370 The terms of this sequence begin a chain of harmonic numbers of a record length. In each chain, each member, except the first, is generated from its predecessor by multiplying it by a prime that does not divide it.
%C A335370 The corresponding record values of k are 0, 1, 2, 3, 4, 6, 7, 8, ...
%C A335370 The list of primes or their order may not be unique.
%e A335370 28 is the least harmonic number with one prime, p = 5, such that 28*p = 140 is also a harmonic number.
%e A335370 1638 is the least harmonic number with 2 primes, 5 and 29, such that 1638*5 = 8190 and 1638*5*29 = 237510 are also harmonic numbers.
%e A335370 .
%e A335370 n a(n) k primes p_i, i = 1..k number of permutations
%e A335370 -------------------------------------------------------------------------------
%e A335370 1 1 0 - -
%e A335370 2 28 1 5 1
%e A335370 3 1638 2 5, 29 1
%e A335370 4 6200 3 19, 37, 73 1
%e A335370 5 2457000 4 11, 19, 37, 73 4
%e A335370 6 4713984 6 5, 7, 13, 19, 37, 73 15
%e A335370 5, 7, 19, 37, 73, 1021 5
%e A335370 7 1381161600 7 11, 19, 37, 43, 73, 6277, 12553 10
%e A335370 11, 19, 37, 43, 3181, 6361, 12721 6
%e A335370 8 10200236032 8 3, 5, 79, 157, 313, 1877, 7507, 15013 5
%t A335370 harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {_, _}][[;; , 2]]; harMean[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; harmGen[n_] := Module[{d = Divisors[harMean[n]]}, n * Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &]]; harmGens[s_] := Union @ Flatten[harmGen /@ s]; lenmax = -1; seq = {}; Do[len = -3 + Length @ FixedPointList[harmGens, {harmNums[[k]]}]; If[len > lenmax, lenmax = len; AppendTo[seq, harmNums[[k]]]], {k, 1, Length[harmNums]}]; seq
%Y A335370 Cf. A000005, A000203, A001599, A099377, A099378, A335368, A335369, A335371.
%K A335370 nonn,more
%O A335370 1,2
%A A335370 _Amiram Eldar_, Jun 03 2020