A374778 Denominator of the mean abundancy index of the divisors of n.

1, 4, 6, 12, 10, 24, 14, 32, 27, 8, 22, 72, 26, 56, 60, 80, 34, 54, 38, 120, 4, 88, 46, 192, 75, 104, 54, 56, 58, 48, 62, 64, 132, 136, 28, 162, 74, 152, 52, 320, 82, 16, 86, 264, 135, 184, 94, 160, 49, 30, 204, 104, 106, 216, 20, 64, 76, 232, 118, 720, 122, 248

Offset: 1

Author

Amiram Eldar, Jul 19 2024

Keywords

• nonn
• easy
• frac

Examples

			For n = 2, n has 2 divisors, 1 and 2. Their abundancy indices are sigma(1)/1 = 1 and sigma(2)/2 = 3/2, and their mean abundancy index is (1 + 3/2)/2 = 5/4. Therefore a(2) = denominator(5/4) = 4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A374777 (numerators).

Programs

Mathematica

f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

PARI

a(n) = {my(f = factor(n), p, e); denominator(prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2)));}