

A056672


Number of unitary and squarefree divisors of n! Also, number of divisors of the special squarefree part of n!, A055773(n).


1



1, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 8, 4, 4, 4, 8, 8, 16, 16, 16, 8, 16, 16, 16, 8, 8, 8, 16, 16, 32, 32, 32, 16, 16, 16, 32, 16, 16, 16, 32, 32, 64, 64, 64, 32, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 64, 128, 128, 256, 128, 128, 128, 128, 128, 256, 256, 256, 256
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OFFSET

1,2


COMMENTS

The divisor d=1 is counted here as being free of prime divisors and also unitary.


LINKS

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


FORMULA

a(n) = A000005(A055231(n!)) = A000005(A007913(n!)/A055229(n!)) = a(n) = A000005(A055773(n)).


EXAMPLE

n=11: 11! = 2*2*2*2*2*2*2*2*3*3*3*3*5*5*7*11, has 540 divisors, 32 are unitary and 32 are squarefree. Only 4 divisors, {1,7,11,77} have both properties, so a(11)=4.


MATHEMATICA

rad[n_] := Times @@ First /@ FactorInteger[n]; p[n_] := Denominator[n/rad[n]^2]; a[n_] := DivisorSigma[0, p[n!]]; Array[a, 70] (* Amiram Eldar, Sep 22 2019 *)


PROG

(PARI) a(n) = my(f=n!); sumdiv(f, d, issquarefree(d) && (gcd(d, f/d) == 1)); \\ Michel Marcus, Sep 05 2017


CROSSREFS

Cf. A000005, A000142, A007913, A055229, A055231, A055773.
Sequence in context: A336409 A010694 A111737 * A037201 A128886 A031883
Adjacent sequences: A056669 A056670 A056671 * A056673 A056674 A056675


KEYWORD

nonn


AUTHOR

Labos Elemer, Aug 10 2000


STATUS

approved



