%I #11 Jan 30 2024 04:07:32

%S 1,3,3,3,3,9,3,4,3,9,3,9,3,9,9,5,3,9,3,9,9,9,3,12,3,9,4,9,3,27,3,6,9,

%T 9,9,9,3,9,9,12,3,27,3,9,9,9,3,15,3,9,9,9,3,12,9,12,9,9,3,27,3,9,9,7,

%U 9,27,3,9,9,27,3,12,3,9,9,9,9,27,3,15,5,9,3

%N The number of divisors of the smallest powerful number that is a multiple of n.

%H Amiram Eldar, <a href="/A369716/b369716.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A369716/a369716.jpg">Graph - the asymptotic ratio (100000 terms)</a>

%F a(n) = A000005(A197863(n)).

%F Multiplicative with a(p) = 3 and a(p^e) = e+1 for e >= 2.

%F a(n) >= A000005(n), with equality if and only if n is powerful (A001694).

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

%F From _Vaclav Kotesovec_, Jan 30 2024: (Start)

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

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

%F Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2), where

%F f(1) = Product_{primes p} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.33718787379158997196169281615215824494915412775816393888028828465611936...,

%F f'(1) = f(1) * Sum_{primes p} (6*p^2 - 9*p + 4) * log(p) / (p^4 - 3*p^2 + 3*p - 1) = f(1) * 2.35603132119230949914708478515883136510141335620960622673206366...,

%F f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-p*(12*p^5 - 27*p^4 + 16*p^3 + 9*p^2 - 12*p + 3) * log(p)^2 / (p^4 - 3*p^2 + 3*p - 1)^2) = f'(1)^2/f(1) + f(1) * (-7.3049026768735124341194605967271037971153161932236518820258070165876...),

%F gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

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

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

%Y Cf. A000005, A001694, A197863, A369717, A369718, A369719.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Jan 30 2024