A340596 Number of co-balanced factorizations of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1

Offset

1,12

Comments

We define a factorization of n into factors > 1 to be co-balanced if it has exactly A001221(n) factors.

Links

Table of n, a(n) for n=1..87.

Example

The a(n) co-balanced factorizations for n = 12, 24, 36, 72, 120, 144, 180:
  2*6    3*8     4*9     8*9     3*5*8     2*72     4*5*9
  3*4    4*6     6*6     2*36    4*5*6     3*48     5*6*6
         2*12    2*18    3*24    2*2*30    4*36     2*2*45
                 3*12    4*18    2*3*20    6*24     2*3*30
                         6*12    2*4*15    8*18     2*5*18
                                 2*5*12    9*16     2*6*15
                                 2*6*10    12*12    2*9*10
                                 3*4*10             3*3*20
                                                    3*4*15
                                                    3*5*12
                                                    3*6*10

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[#]==PrimeNu[n]&]], {n, 100}]

Crossrefs

Positions of terms > 1 are A126706.

Positions of 1's are A303554.

The version for unlabeled multiset partitions is A319616.

The alt-balanced version is A340599.

The balanced version is A340653.

The cross-balanced version is A340654.

The twice-balanced version is A340655.

A001055 counts factorizations.

A045778 counts strict factorizations.

A316439 counts factorizations by product and length.

Other balance-related sequences:

- A010054 counts balanced strict partitions.

- A047993 counts balanced partitions.

- A098124 counts balanced compositions.

- A106529 lists Heinz numbers of balanced partitions.

- A340597 lists numbers with an alt-balanced factorization.

- A340598 counts balanced set partitions.

- A340600 counts unlabeled balanced multiset partitions.

Cf. A003963, A006141, A050320, A112798, A117409, A324518, A339846, A339890, A340607, A340656, A340657.

Sequence in context: A327528 A264857 A370645 * A340654 A303837 A286520

Adjacent sequences: A340593 A340594 A340595 * A340597 A340598 A340599

Keyword

nonn

Author

Gus Wiseman, Jan 15 2021

Status

approved