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

1,8

Links

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

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... .

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A368172 A359594 A368331 * A204160 A360165 A336870

Adjacent sequences: A366142 A366143 A366144 * A366146 A366147 A366148

Keyword

nonn,easy,mult

Author

Amiram Eldar, Oct 01 2023

Status

approved