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A365550
The number of square coreful divisors of n.
1
1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0
OFFSET
1,16
COMMENTS
First differs from A188585 at n = 64.
A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n.
LINKS
FORMULA
Multiplicative with a(p^e) = floor(e/2).
a(n) > 0 if and only if n is a powerful number (A001694).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).
EXAMPLE
a(16) = 2 since the coreful divisors of 16 are {2, 4, 8, 16}, and 2 of them, 4 and 16, are squares.
MATHEMATICA
f[p_, e_] := Floor[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> x\2, factor(n)[, 2]));
CROSSREFS
Cf. A001694, A005361 (number of coreful divisors), A046951 (number of square divisors), A325837.
Sequence in context: A245196 A259362 A303553 * A188585 A294068 A339553
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 08 2023
STATUS
approved