OFFSET
1,2
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.
EXAMPLE
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
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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
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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Jun 03 2020
STATUS
approved