A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 2, 4, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4

Offset

1,2

Comments

First differs from A000005 at n = 25.

The sum of these divisors is A365209(n).

Links

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

Formula

Multiplicative with a(p^e) = e+1 if p = 2 or 3, and a(p^e) = 2 for a prime p >= 5.

a(n) <= A000005(n), with equality if and only if n is not divisible by a square of a prime >= 5.

a(n) >= A034444(n), with equality if and only if n is neither divisible by 4 nor by 9.

a(n) = A000005(A065331(n)) * A034444(A065330(n)).

Dirichlet g.f.: (4^s/(4^s-1)) * (9^s/(9^s-1)) * zeta(s)^2/zeta(2*s).

Sum_{k==1..n} a(k) ~ (9/Pi^2)*n*(log(n) + 2*gamma - 2*log(2)/3 - log(3)/4 - 2*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620).

Mathematica

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

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] <= 3, f[i, 2]+1, 2)); }

Crossrefs

Cf. A000005, A003586, A034444, A065330, A065331, A365209.

Cf. A001620, A013661, A073002.

Sequence in context: A365680 A325560 A318412 * A322986 A335519 A167447

Adjacent sequences: A365205 A365206 A365207 * A365209 A365210 A365211

Keyword

nonn,easy,mult

Author

Amiram Eldar, Aug 26 2023

Status

approved