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A361782
Numerators of the harmonic means of the bi-unitary divisors of the positive integers.
5
1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 64, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 64, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 32, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 384, 65, 11, 67, 136
OFFSET
1,2
LINKS
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.
FORMULA
a(n) = numerator(n*A286324(n)/A188999(n)).
f(n) = a(n)/A361783(n) is multiplicative with f(p^e) = (e+1)*(p-1)/(p^(e+1)-1) if e is odd, and e/((p^(e+1)-1)/(p-1) - p^(e/2)) if e is even.
EXAMPLE
Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...
MATHEMATICA
f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n), p, e); numerator(n * prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))))); }
CROSSREFS
Cf. A188999, A222266, A286324, A361783 (denominators).
Similar sequences: A099377, A103339, A361316.
Sequence in context: A097672 A103339 A361316 * A353990 A092383 A227242
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar, Mar 24 2023
STATUS
approved