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
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
Programs
-
Mathematica
f[p_, e_] := e + 1 + Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
-
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
Formula
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...,
Comments