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

1,4

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

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.

Mathematica

f[p_, e_] := (p^e - 1)/(p-1) - 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~, (f[i, 1]^f[i, 2] - 1)/(f[i, 1] - 1) - 1); }

Crossrefs

Cf. A001694, A307958, A361430.

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

Sequence in context: A309746 A094315 A212148 * A358622 A336563 A048146

Adjacent sequences: A364985 A364986 A364987 * A364989 A364990 A364991

Keyword

nonn,mult

Author

Amiram Eldar, Aug 15 2023

Status

approved