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

Author

Geoffrey Critzer, Jul 09 2015

Keywords

• nonn

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

Examples

			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
		

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

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

Programs

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 *)

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

Formula

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

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

Extensions

Incorrect comment removed by Antti Karttunen, Jun 11 2022