A361782 Numerators of the harmonic means of the bi-unitary divisors of the positive integers.

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

Author

Amiram Eldar, Mar 24 2023

Keywords

• nonn
• frac

Examples

			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, ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.

Crossrefs

Cf. A188999, A222266, A286324, A361783 (denominators).

Similar sequences: A099377, A103339, A361316.

Programs

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]

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))))); }

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.