A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

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

Offset: 1

Author

Amiram Eldar, Jun 16 2025

Keywords

• nonn
• easy
• mult

Comments

The sum of these divisors is A385045(n), and the largest of them is A065330(n).

Links

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

Crossrefs

The unitary analog of A035218.

Cf. A007310, A034444, A065330, A385045.

Cf. A001620, A013661, A306016.

The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), A385042 (exponentially 2^n), this sequence (5-rough).

Programs

Mathematica

f[p_, e_] := If[p <= 3, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

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

Formula

Multiplicative with a(p^e) = 1 if p <= 3, and 2 if p >= 5.

a(n) = A034444(n)/A382488(n).

a(n) <= A034444(n), with equality if and only if n is 5-rough.

a(n) <= A035218(n).

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

Sum_{k=1..n} a(k) ~ (n / (2 * zeta(2))) *(log(n) + 2*gamma - 1 + log(2)/3 + log(3)/4 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).