A374902 Numerator of the mean number of divisors of the divisors of n.

1, 3, 3, 2, 3, 9, 3, 5, 2, 9, 3, 3, 3, 9, 9, 3, 3, 3, 3, 3, 9, 9, 3, 15, 2, 9, 5, 3, 3, 27, 3, 7, 9, 9, 9, 4, 3, 9, 9, 15, 3, 27, 3, 3, 3, 9, 3, 9, 2, 3, 9, 3, 3, 15, 9, 15, 9, 9, 3, 9, 3, 9, 3, 4, 9, 27, 3, 3, 9, 27, 3, 5, 3, 9, 3, 3, 9, 27, 3, 9, 3, 9, 3, 9, 9, 9, 9

Offset: 1

Author

Amiram Eldar, Jul 23 2024

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin: 1, 3/2, 3/2, 2, 3/2, 9/4, 3/2, 5/2, 2, 9/4, ...
For n = 2, n has 2 divisors, 1 and 2. Their numbers of divisors are tau(1) = 1 and tau(2) = 2, and their mean number of divisors is (1 + 2)/2 = 3/2. Therefore a(2) = numerator(3/2) = 3.
		

Links

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

Crossrefs

Cf. A000005, A007425, A374903 (denominators).

Programs

Mathematica

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

PARI

a(n) = numerator(vecprod(apply(x -> x/2 +1, factor(n)[, 2])));

Formula

Let f(n) = a(n)/A374903(n). Then:

f(n) = (Sum_{d|n} tau(d)) / tau(n), where tau(n) is the number of divisors of n (A000005).

f(n) = A007425(n)/A000005(n).

f(n) is multiplicative with f(p^e) = 1 + e/2.

Dirichlet g.f. of f(n): zeta(s)^2 * Product_{p prime} (1 - 1/(2*p^s)).