A382488 The number of unitary 3-smooth divisors of n.

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

Offset: 1

Author

Amiram Eldar, Mar 29 2025

Keywords

• nonn
• easy
• mult

Comments

Period 6: repeat [1, 2, 2, 2, 1, 4].

Decimal expansion of 407380/333333.

Continued fraction expansion of 10/(6 + sqrt(66)) (with offset 0).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Crossrefs

Cf. A005117, A003586, A034444, A065331, A072078, A181982, A382487.

The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), this sequence (k = 2), A382489 (k = 3).

Programs

Mathematica

Table[{1, 2, 2, 2, 1, 4}, {12}] // Flatten

PARI

a(n) = [1, 2, 2, 2, 1, 4][(n-1) % 6 + 1];

Formula

Multiplicative with a(p^e) = 2 if p <= 3, and 1 otherwise.

a(n) = A034444(A065331(n)).

a(n) = A034444(n) if and only if n is 3-smooth (A003586).

a(n) = A072078(n) if and only if n is squarefree (A005117).

a(n) = abs(A181982(n+9)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.

G.f.: -(4*x^6 + x^5 + 2*x^4 + 2*x^3 +2*x^2 + x)/(x^6 - 1).

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