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

%I A374902 #7 Jul 23 2024 20:46:32
%S A374902 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,
%T A374902 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,
%U A374902 27,3,3,9,27,3,5,3,9,3,3,9,27,3,9,3,9,3,9,9,9,9
%N A374902 Numerator of the mean number of divisors of the divisors of n.
%H A374902 Amiram Eldar, <a href="/A374902/b374902.txt">Table of n, a(n) for n = 1..10000</a>
%F A374902 Let f(n) = a(n)/A374903(n). Then:
%F A374902 f(n) = (Sum_{d|n} tau(d)) / tau(n), where tau(n) is the number of divisors of n (A000005).
%F A374902 f(n) = A007425(n)/A000005(n).
%F A374902 f(n) is multiplicative with f(p^e) = 1 + e/2.
%F A374902 Dirichlet g.f. of f(n): zeta(s)^2 * Product_{p prime} (1 - 1/(2*p^s)).
%e A374902 Fractions begin: 1, 3/2, 3/2, 2, 3/2, 9/4, 3/2, 5/2, 2, 9/4, ...
%e A374902 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.
%t A374902 f[p_, e_] := (e + 2)/2; a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]
%o A374902 (PARI) a(n) = numerator(vecprod(apply(x -> x/2 +1, factor(n)[, 2])));
%Y A374902 Cf. A000005, A007425, A374903 (denominators).
%K A374902 nonn,easy,frac
%O A374902 1,2
%A A374902 _Amiram Eldar_, Jul 23 2024