A365345 The number of divisors of the smallest square divisible by n.

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

Offset

1,2

Comments

The sum of these divisors is A365346(n).

The number of divisors of the square root of the smallest square divisible by n is A322483(n).

Links

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

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

Formula

a(n) = A000005(A053143(n)).

Multiplicative with a(p^e) = e + 1 + (e mod 2).

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

From Vaclav Kotesovec, Sep 05 2023: (Start)

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

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

Sum_{k=1..n} a(k) ~ f(1) * Pi^2 * n / 6 * (log(n)^2/2 + (3*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)) * log(n) + 1 - 3*gamma + 3*gamma^2 - 3*sg1 + (3*gamma - 1)*12*zeta'(2)/Pi^2 + 12*zeta''(2)/Pi^2 + (12*zeta'(2)/Pi^2 + 3*gamma - 1)*f'(1)/f(1) + f''(1)/(2*f(1))), where

f(1) = Product_{primes p} (1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,

f'(1) = f(1) * Sum_{primes p} 4*(2*p - 1) * log(p) / (1 - 3*p + p^2 + p^3) = 0.7343690473711153863995729489689746152413988981744946512300478410459132782...

f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} 4*p*(-1 + 2*p + p^2 - 4*p^3) * log(p)^2 / (1 - 3*p + p^2 + p^3)^2 = 0.1829055032494906699795154632343894745397324334876662084674149254022564139...,

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

Mathematica

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

Prog

(PARI) a(n) = vecprod(apply(x -> x + 1 + x%2, factor(n)[, 2]));
(PARI) a(n) = numdiv(n*core(n)); \\ Michel Marcus, Sep 02 2023

Crossrefs

Cf. A000005, A053143, A322483, A365346.

Sequence in context: A074816 A203564 A369716 * A375230 A111575 A161836

Adjacent sequences: A365342 A365343 A365344 * A365346 A365347 A365348

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 02 2023

Status

approved