

A259936


Number of ways to express the integer n as a product of its unitary divisors (A034444).


18



1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 5, 2, 2, 2, 2, 1, 5, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 5
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OFFSET

1,6


COMMENTS

Equivalently, a(n) is the number of ways to express the cyclic group Z_n as a direct sum of its Hall subgroups. A Hall subgroup of a finite group G is a subgroup whose order is coprime to its index.
a(n) is the number of ways to partition the set of distinct prime factors of n.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Wikipedia, Hall subgroup
Index entries for sequences computed from exponents in factorization of n


FORMULA

a(n) = A000110(A001221(n)).


EXAMPLE

a(60) = 5 because we have: 60 = 4*3*5 = 4*15 = 3*20 = 5*12.
For n = 36, its unitary divisors are 1, 4, 9, 36. From these we obtain 36 either as 1*36 or 4*9, thus a(36) = 2.  Antti Karttunen, Oct 21 2017


MAPLE

map(combinat:bell @ nops @ numtheory:factorset, [$1..100]); # Robert Israel, Jul 09 2015


MATHEMATICA

Table[BellB[PrimeNu[n]], {n, 1, 75}]


PROG

(PARI) a(n) = my(t=omega(n), x='x, m=contfracpnqn(matrix(2, t\2, y, z, if( y==1, z*x^2, 1  (z+1)*x)))); polcoeff(1/(1  x + m[2, 1]/m[1, 1]) + O(x^(t+1)), t) \\ Charles R Greathouse IV, Jun 30 2017


CROSSREFS

Cf. A000110, A001055, A001221, A034444, A089233, A258466, A281116, A285572.
Differs from A050320 for the first time at n=36.
Sequence in context: A318369 A007875 A323437 * A050320 A294893 A121382
Adjacent sequences: A259933 A259934 A259935 * A259937 A259938 A259939


KEYWORD

nonn


AUTHOR

Geoffrey Critzer, Jul 09 2015


STATUS

approved



