A327399 Number of factorizations of n that are constant or whose distinct factors are pairwise coprime.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 6, 1, 2, 2, 3, 1, 5, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 7, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 5, 1, 2, 3, 3, 2, 5, 1, 4, 3, 2, 1, 7, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Sep 22 2019

Keywords

• nonn
• changed

Comments

First differs from A327400 at A327400(24) = 4, a(24) = 3.

From Jianing Song, Jun 09 2025: (Start)

Let n = (p_1)^(e_1) * ... * (p_r)^(e_r), then a(n) is the number of partitions of the multiset formed by e_1 1's, e_2 2's, ..., e_r r's such that each pair of parts is either equal or nonintersecting. Let's call such a partition a (e_1,...,e_r)-partition of {1,2,...,r}.

Note that every (e_1,...,e_r)-partition has a base partition by removing duplicates of parts and elements in each part (e.g., {{1,2,2},{1,2,2},{3,3},{4}} -> {{1,2},{3},{4}}), and the base partition is itself a partition on {1,2,...,r}. Since the number of partitions into identical parts of the multiset formed by e_{i_1} (i_1)'s, ..., e_{i_k} (i_k)'s is d(gcd(e_{i_1},...,e_{i_k})), where d = A000005, the number of (e_1,...,e_r)-partitions having base partition P of {1,2,...,r} is Product_{S in P} d(gcd_{i in S} (e_i)). As a result, the number (e_1,...,e_r)-partitions is Sum_{P is a partition of {1,2,...,r}} Product_{S in P} d(gcd_{i in S} (e_i)).

Examples:

# of e_1-partitions = d(e_1);

# of (e_1,e_2)-partitions = d(gcd(e_1,e_2)) + d(e_1)*d(e_2);

# of (e_1,e_2,e_3)-partitions = d(gcd(e_1,e_2,e_3)) + d(gcd(e_1,e_2))*d(e_3) + d(gcd(e_1,e_3))*d(e_2) + d(gcd(e_2,e_3))*d(e_1) + d(e_1)*d(e_2)*d(e_3);

# of (e_1,e_2,e_3,e_4)-partitions = d(gcd(e_1,e_2,e_3,e_4)) + (d(gcd(e_1,e_2,e_3))*d(e_4) + ...) + (d(gcd(e_1,e_2))*d(gcd(e_3,e_4)) + ...) + (d(gcd(e_1,e_2))*d(e_3)*d(e_4) + ...) + d(e_1)*d(e_2)*d(e_3)*d(e_4).

(End)

Examples

			The a(90) = 7 factorizations together with the corresponding multiset partitions of {1,2,2,3}:
  (2*3*3*5)  {{1},{2},{2},{3}}
  (2*5*9)    {{1},{3},{2,2}}
  (2*45)     {{1},{2,2,3}}
  (3*3*10)   {{2},{2},{1,3}}
  (5*18)     {{3},{1,2,2}}
  (9*10)     {{2,2},{1,3}}
  (90)       {{1,2,2,3}}
		

Links

• Gus Wiseman, Sequences counting and encoding certain classes of multisets

Crossrefs

Constant factorizations are A089723.

Partitions whose distinct parts are pairwise coprime are A304709.

Factorizations that are constant or relatively prime are A327400.

See link for additional cross-references.

Cf. A007359, A050320, A051424, A281116, A302569, A302696, A304711.

Programs

Mathematica

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[Union[#]]==1||CoprimeQ@@Union[#]&]],{n,100}]

Formula

a(n) = A327695(n) + A089723(n).