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A364988
a(n) is the sum of coreful divisors d of n such that n/d is also a coreful divisor.
1
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
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
Similar sequences: A000203, A057723 (sum of coreful divisors).
Sequence in context: A309746 A094315 A212148 * A358622 A336563 A048146
KEYWORD
nonn,mult
AUTHOR
Amiram Eldar, Aug 15 2023
STATUS
approved