A382489 The number of unitary 5-smooth divisors of n.

1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2

Offset: 1

Author

Amiram Eldar, Mar 29 2025

Keywords

• nonn
• easy
• mult

Comments

Period 30: repeat [1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8].

In general, the sequence of the number of unitary prime(k)-smooth divisors of n, for k >= 1, is periodic with period A002110(k).

Decimal expansion of 135804580460138015713571358020/111111111111111111111111111111.

Continued fraction expansion of 808690/(525316 + sqrt(382161348866)) (with offset 0).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A002110, A005117, A034444, A051037, A054640, A236435, A236436, A355582, A355583.

The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), A382488 (k = 2), this sequence (k = 3).

Programs

Mathematica

a[n_] := Product[If[Divisible[n, p], 2, 1], {p, {2, 3, 5}}]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> !((n % 30) % x) + 1, [2, 3, 5]))

Formula

Multiplicative with a(p^e) = 2 if p <= 5, and 1 otherwise.

a(n) = A034444(A355582(n)).

a(n) = A034444(n) if and only if n is 5-smooth (A051037).

a(n) = A355583(n) if and only if n is squarefree (A005117).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 12/5.

In general, the asymptotic mean of the number of unitary prime(k)-smooth divisors of n is A054640(k)/A002110(k) = A236435(k)/A236436(k).

Dirichlet g.f.: (1 + 1/2^s) * (1 + 1/3^s) * (1 + 1/5^s) * zeta(s).

In general, Dirichlet g.f. of the number of unitary prime(k)-smooth divisors of n is zeta(s) * Product_{p prime <= prime(k)} (1 + 1/p^s).