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A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444). 23

%I

%S 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,

%T 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,

%U 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

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

%C 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.

%C a(n) is the number of ways to partition the set of distinct prime factors of n.

%C Also the number of singleton or pairwise coprime factorizations of n. - _Gus Wiseman_, Sep 24 2019

%H Alois P. Heinz, <a href="/A259936/b259936.txt">Table of n, a(n) for n = 1..20000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hall_subgroup">Hall subgroup</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

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

%F a(n > 1) = A327517(n) + 1. - _Gus Wiseman_, Sep 24 2019

%e a(60) = 5 because we have: 60 = 4*3*5 = 4*15 = 3*20 = 5*12.

%e 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

%p map(combinat:-bell @ nops @ numtheory:-factorset, [$1..100]); # _Robert Israel_, Jul 09 2015

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

%t (* second program *)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],Length[#]==1||CoprimeQ@@#&]],{n,100}] (* _Gus Wiseman_, Sep 24 2019 *)

%o (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

%Y Cf. A000110, A001055, A001221, A034444, A089233, A258466, A281116, A285572.

%Y Differs from A050320 for the first time at n=36.

%Y Cf. A304716, A302569, A304711, A305079.

%Y Related classes of factorizations:

%Y - No conditions: A001055

%Y - Strict: A045778

%Y - Constant: A089723

%Y - Distinct multiplicities: A255231

%Y - Singleton or coprime: A259936

%Y - Relatively prime: A281116

%Y - Aperiodic: A303386

%Y - Stable (indivisible): A305149

%Y - Connected: A305193

%Y - Strict relatively prime: A318721

%Y - Uniform: A319269

%Y - Intersecting: A319786

%Y - Constant or distinct factors coprime: A327399

%Y - Constant or relatively prime: A327400

%Y - Coprime: A327517

%Y - Not relatively prime: A327658

%Y - Distinct factors coprime: A327695

%K nonn

%O 1,6

%A _Geoffrey Critzer_, Jul 09 2015

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Last modified August 1 00:13 EDT 2021. Contains 346377 sequences. (Running on oeis4.)