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

1,2

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).

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).

Mathematica

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

Prog

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

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).

Sequence in context: A216322 A335383 A125914 * A086668 A092904 A231883

Adjacent sequences: A382486 A382487 A382488 * A382490 A382491 A382492

Keyword

nonn,easy,mult

Author

Amiram Eldar, Mar 29 2025

Status

approved