A364988 a(n) is the sum of coreful divisors d of n such that n/d is also a coreful divisor.

1, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 12, 0, 0, 0, 0, 30, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 62, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 39, 0, 0, 0, 0

Offset: 1

Author

Amiram Eldar, Aug 15 2023

Keywords

• nonn
• mult

Comments

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).

The number of these divisors is A361430(n).

Links

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

Crossrefs

Cf. A001694, A307958, A361430.

Similar sequences: A000203, A057723 (sum of coreful divisors).

Programs

Mathematica

f[p_, e_] := (p^e - 1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^f[i,2] - 1)/(f[i,1] - 1) - 1);}

Formula

Multiplicative with a(p^e) = (p^e - 1)/(p-1) - 1.

Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + (2*p - p^s*(p+1))/p^(2*s)).

a(n) > 0 if and only if n is powerful (A001694).

a(n) <= n with equality only when n = 1.

a(p^2) = p for a prime p.