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A335370
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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.
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3
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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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
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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