cp's OEIS Frontend

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

%I A259936 #30 Jun 11 2022 14:03:47
%S A259936 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 A259936 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 A259936 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 A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).
%C A259936 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 A259936 a(n) is the number of ways to partition the set of distinct prime factors of n.
%C A259936 Also the number of singleton or pairwise coprime factorizations of n. - _Gus Wiseman_, Sep 24 2019
%H A259936 Alois P. Heinz, <a href="/A259936/b259936.txt">Table of n, a(n) for n = 1..20000</a>
%H A259936 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hall_subgroup">Hall subgroup</a>
%H A259936 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A259936 a(n) = A000110(A001221(n)).
%F A259936 a(n > 1) = A327517(n) + 1. - _Gus Wiseman_, Sep 24 2019
%e A259936 a(60) = 5 because we have: 60 = 4*3*5 = 4*15 = 3*20 = 5*12.
%e A259936 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 A259936 map(combinat:-bell @ nops @ numtheory:-factorset, [$1..100]); # _Robert Israel_, Jul 09 2015
%t A259936 Table[BellB[PrimeNu[n]], {n, 1, 75}]
%t A259936 (* second program *)
%t A259936 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A259936 Table[Length[Select[facs[n],Length[#]==1||CoprimeQ@@#&]],{n,100}] (* _Gus Wiseman_, Sep 24 2019 *)
%o A259936 (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 A259936 Cf. A000110, A001055, A001221, A034444, A089233, A258466, A281116, A285572.
%Y A259936 Differs from A050320 for the first time at n=36.
%Y A259936 Differs from A354870 for the first time at n=210, where a(210) = 15, while A354870(210) = 12.
%Y A259936 Cf. A304716, A302569, A304711, A305079.
%Y A259936 Related classes of factorizations:
%Y A259936 - No conditions: A001055
%Y A259936 - Strict: A045778
%Y A259936 - Constant: A089723
%Y A259936 - Distinct multiplicities: A255231
%Y A259936 - Singleton or coprime: A259936
%Y A259936 - Relatively prime: A281116
%Y A259936 - Aperiodic: A303386
%Y A259936 - Stable (indivisible): A305149
%Y A259936 - Connected: A305193
%Y A259936 - Strict relatively prime: A318721
%Y A259936 - Uniform: A319269
%Y A259936 - Intersecting: A319786
%Y A259936 - Constant or distinct factors coprime: A327399
%Y A259936 - Constant or relatively prime: A327400
%Y A259936 - Coprime: A327517
%Y A259936 - Not relatively prime: A327658
%Y A259936 - Distinct factors coprime: A327695
%K A259936 nonn
%O A259936 1,6
%A A259936 _Geoffrey Critzer_, Jul 09 2015
%E A259936 Incorrect comment removed by _Antti Karttunen_, Jun 11 2022