A366145 The number of divisors of the largest divisor of n that is a cubefull number (A036966).

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Oct 01 2023

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000005, A036966, A357669, A360540, A366076, A366146, A366147.

Programs

Mathematica

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

PARI

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

Formula

a(n) = A000005(A360540(n)).

a(n) = A000005(n)/A366147(n).

a(n) >= 1, with equality if and only if n is cubefree (A004709).

a(n) <= A000005(n), with equality if and only if n is cubefull (A036966).

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

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 3/p^3 + 1/p^4 - 2/p^5) = 1.76434793373691907811... .