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A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444). 7
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 (list; graph; refs; listen; history; text; internal format)
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: A064372 A096825 A007875 * A050320 A294893 A121382

Adjacent sequences:  A259933 A259934 A259935 * A259937 A259938 A259939

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Jul 09 2015

STATUS

approved

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Last modified February 19 16:28 EST 2018. Contains 299356 sequences. (Running on oeis4.)