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A259936
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Number of ways to express the integer n as a product of its unitary divisors (A034444).
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23
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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
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OFFSET
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1,6
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COMMENTS
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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
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LINKS
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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
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FORMULA
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a(n) = A000110(A001221(n)).
a(n > 1) = A327517(n) + 1. - Gus Wiseman, Sep 24 2019
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EXAMPLE
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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
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MAPLE
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map(combinat:-bell @ nops @ numtheory:-factorset, [$1..100]); # Robert Israel, Jul 09 2015
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000110, A001055, A001221, A034444, A089233, A258466, A281116, A285572.
Differs from A050320 for the first time at n=36.
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 * A050320 A333175 A294893
Adjacent sequences: A259933 A259934 A259935 * A259937 A259938 A259939
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KEYWORD
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nonn
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AUTHOR
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Geoffrey Critzer, Jul 09 2015
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STATUS
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approved
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