A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).

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

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.

Also the number of singleton or pairwise coprime factorizations of n. - Gus Wiseman, Sep 24 2019

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)).

a(n > 1) = A327517(n) + 1. - Gus Wiseman, Sep 24 2019

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}]
(* second program *)
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Length[#]==1||CoprimeQ@@#&]], {n, 100}] (* Gus Wiseman, Sep 24 2019 *)

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.

Differs from A354870 for the first time at n=210, where a(210) = 15, while A354870(210) = 12.

Cf. A304716, A302569, A304711, A305079.

Related classes of factorizations:

- No conditions: A001055

- Strict: A045778

- Constant: A089723

- Distinct multiplicities: A255231

- Singleton or coprime: A259936

- Relatively prime: A281116

- Aperiodic: A303386

- Stable (indivisible): A305149

- Connected: A305193

- Strict relatively prime: A318721

- Uniform: A319269

- Intersecting: A319786

- Constant or distinct factors coprime: A327399

- Constant or relatively prime: A327400

- Coprime: A327517

- Not relatively prime: A327658

- Distinct factors coprime: A327695

Sequence in context: A007875 A323437 A339887 * A354870 A050320 A333175

Adjacent sequences: A259933 A259934 A259935 * A259937 A259938 A259939

Keyword

nonn

Author

Geoffrey Critzer, Jul 09 2015

Extensions

Incorrect comment removed by Antti Karttunen, Jun 11 2022

Status

approved