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

Author

Amiram Eldar, Aug 26 2023

Keywords

• nonn
• easy
• mult

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

Crossrefs

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

Cf. A001620, A013661, A073002.

Programs

Mathematica

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

PARI

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

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).