A342086 Number of strict factorizations of divisors of n.

1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 9, 2, 5, 5, 7, 2, 9, 2, 9, 5, 5, 2, 16, 3, 5, 5, 9, 2, 15, 2, 10, 5, 5, 5, 18, 2, 5, 5, 16, 2, 15, 2, 9, 9, 5, 2, 25, 3, 9, 5, 9, 2, 16, 5, 16, 5, 5, 2, 31, 2, 5, 9, 14, 5, 15, 2, 9, 5, 15, 2, 34, 2, 5, 9, 9, 5, 15, 2, 25, 7, 5

Offset

1,2

Comments

A strict factorization of n is a set of distinct positive integers > 1 with product n.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

The a(1) = 1 through a(12) = 9 factorizations:
  ()  ()   ()   ()   ()   ()     ()   ()     ()   ()     ()    ()
      (2)  (3)  (2)  (5)  (2)    (7)  (2)    (3)  (2)    (11)  (2)
                (4)       (3)         (4)    (9)  (5)          (3)
                          (6)         (8)         (10)         (4)
                          (2*3)       (2*4)       (2*5)        (6)
                                                               (12)
                                                               (2*3)
                                                               (2*6)
                                                               (3*4)

Maple

sf1:= proc(n, m)
  local D, d;
  if n = 1 then return 1 fi;
  D:= select(`<`, numtheory:-divisors(n) minus {1}, m);
  add( procname(n/d, d), d= D)
end proc:
sf:= proc(n) option remember; sf1(n, n+1) end proc:f:= proc(n) local d; add(sf(d), d=numtheory:-divisors(n)) end proc:map(f, [$1..100]); # Robert Israel, Mar 10 2021

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Sum[Length[Select[facs[k], UnsameQ@@#&]], {k, Divisors[n]}], {n, 30}]

Crossrefs

A version for partitions is A026906 (strict partitions of 1..n).

A version for partitions is A036469 (strict partitions of 0..n).

A version for partitions is A047966 (strict partitions of divisors).

The non-strict version is A057567.

A000005 counts divisors, with sum A000203.

A000009 counts strict partitions.

A001055 counts factorizations, with strict case A045778.

A001221 counts prime divisors, with sum A001414.

A001222 counts prime-power divisors.

A005117 lists squarefree numbers.

Cf. A001227, A050320, A340101, A340596, A340654, A340655, A340853, A341596, A341673, A341674, A342097.

Sequence in context: A073023 A173754 A180125 * A272209 A326842 A326843

Adjacent sequences: A342083 A342084 A342085 * A342087 A342088 A342089

Keyword

nonn

Author

Gus Wiseman, Mar 05 2021

Status

approved