A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 4, 3, 4, 2, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 3, 6, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 6, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset

1,2

Comments

First differs from A034444 at n = 25.

The sum of these divisors is A365211(n).

Links

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

Formula

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

a(n) <= A000005(n), with equality if and only if n is neither divisible by 4 nor by 9.

a(n) >= A034444(n), with equality if and only if n is not divisible by a square of a prime >= 5.

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

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

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

Mathematica

f[p_, e_] := If[p <= 3 , 2, e + 1]; 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, 2, f[i, 2]+1)); }

Crossrefs

Cf. A000005, A007310, A035218, A065330, A065331, A069733, A365211.

Cf. A001620, A013661, A073002.

Sequence in context: A158522 A034444 A365491 * A365488 A365171 A369310

Adjacent sequences: A365207 A365208 A365209 * A365211 A365212 A365213

Keyword

nonn,easy,mult

Author

Amiram Eldar, Aug 26 2023

Status

approved