A369716 The number of divisors of the smallest powerful number that is a multiple of n.

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, 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, 9, 27, 3, 9, 9, 27, 3, 12, 3, 9, 9, 9, 9, 27, 3, 15, 5, 9, 3

Offset

1,2

Links

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

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Formula

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

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

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

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

From Vaclav Kotesovec, Jan 30 2024: (Start)

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

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

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

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''(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...),

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A184099 A074816 A203564 * A365345 A375230 A111575

Adjacent sequences: A369713 A369714 A369715 * A369717 A369718 A369719

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 30 2024

Status

approved