A366147 - OEIS

%I #10 Oct 02 2023 01:44:12

%S 1,2,2,3,2,4,2,1,3,4,2,6,2,4,4,1,2,6,2,6,4,4,2,2,3,4,1,6,2,8,2,1,4,4,

%T 4,9,2,4,4,2,2,8,2,6,6,4,2,2,3,6,4,6,2,2,4,2,4,4,2,12,2,4,6,1,4,8,2,6,

%U 4,8,2,3,2,4,6,6,4,8,2,2,1,4,2,12,4,4,4

%N The number of divisors of the cubefree part of n (A360539).

%H Amiram Eldar, <a href="/A366147/b366147.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(A360539(n)).

%F a(n) = A000005(n)/A366145(n).

%F Multiplicative with a(p^e) = e+1 if e <= 2, and 1 otherwise.

%F a(n) >= 1, with equality if and only if n is cubefull (A036966).

%F a(n) <= A000005(n), with equality if and only if n is cubefree (A004709).

%F Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s) - 2/p^(3*s)).

%F From _Vaclav Kotesovec_, Oct 01 2023: (Start)

%F Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 3/p^(3*s) + 2/p^(4*s)).

%F Let f(s) = Product_{p prime} (1 - 3/p^(3*s) + 2/p^(4*s)).

%F Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where

%F f(1) = Product_{p prime} (1 - 3/p^3 + 2/p^4) = 0.66219033176371496870504912254207846719824904470940603905284774924086...,

%F f'(1) = f(1) * Sum_{p prime} (9*p - 8) * log(p) / (p^4 - 3*p + 2) = f(1) * 1.04316863044761953555286128194165251303791613504188623828521117799260...

%F and gamma is the Euler-Mascheroni constant A001620. (End)

%t f[p_, e_] := If[e < 3, e+1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecprod(apply(x -> if(x < 3, x+1, 1), factor(n)[, 2]));

%Y Cf. A000005, A004709, A036966, A056671, A360539, A366077, A366078, A366145, A366148.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Oct 01 2023