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.

1, 28, 1638, 6200, 2457000, 4713984, 1381161600, 10200236032

Offset: 1

Author

Amiram Eldar, Jun 03 2020

Keywords

• nonn
• more

Comments

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).

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.

The corresponding record values of k are 0, 1, 2, 3, 4, 6, 7, 8, ...

The list of primes or their order may not be unique.

Examples

			28 is the least harmonic number with one prime, p = 5, such that 28*p = 140 is also a harmonic number.
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.
.
n  a(n)         k   primes p_i, i = 1..k                 number of permutations
-------------------------------------------------------------------------------
1  1            0   -                                         -
2  28           1   5                                         1
3  1638         2   5, 29                                     1
4  6200         3   19, 37, 73                                1
5  2457000      4   11, 19, 37, 73                            4
6  4713984      6   5, 7, 13, 19, 37, 73                      15
                    5, 7, 19, 37, 73, 1021                    5
7  1381161600   7   11, 19, 37, 43, 73, 6277, 12553           10
                    11, 19, 37, 43, 3181, 6361, 12721         6
8  10200236032  8   3, 5, 79, 157, 313, 1877, 7507, 15013     5
		

Crossrefs

Cf. A000005, A000203, A001599, A099377, A099378, A335368, A335369, A335371.

Programs

Mathematica

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