A340654 Number of cross-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, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1

Offset

1,12

Comments

We define a factorization of n into factors > 1 to be cross-balanced if either (1) it is empty or (2) the maximum image of A001222 over the factors is A001221(n).

Links

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

Example

The cross-balanced factorizations for n = 12, 24, 36, 72, 144, 240:
  2*6   4*6     4*9     2*4*9     4*4*9       8*30
  3*4   2*2*6   6*6     2*6*6     4*6*6       12*20
        2*3*4   2*2*9   3*4*6     2*2*4*9     5*6*8
                2*3*6   2*2*2*9   2*2*6*6     2*4*30
                3*3*4   2*2*3*6   2*3*4*6     2*6*20
                        2*3*3*4   3*3*4*4     2*8*15
                                  2*2*2*2*9   3*4*20
                                  2*2*2*3*6   3*8*10
                                  2*2*3*3*4   4*5*12
                                              2*10*12
                                              2*3*5*8
                                              2*2*2*30
                                              2*2*3*20
                                              2*2*5*12

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], #=={}||PrimeNu[n]==Max[PrimeOmega/@#]&]], {n, 100}]

Crossrefs

Positions of terms > 1 are A126706.

Positions of 1's are A303554.

The co-balanced version is A340596.

The version for unlabeled multiset partitions is A340651.

The balanced version is A340653.

The twice-balanced version is A340655.

A001055 counts factorizations.

A045778 counts strict factorizations.

A316439 counts factorizations by product and length.

A320655 counts factorizations into semiprimes.

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 have an alt-balanced factorization.

- A340598 counts balanced set partitions.

- A340599 counts alt-balanced factorizations.

- A340652 counts unlabeled twice-balanced multiset partitions.

- A340656 have no twice-balanced factorizations.

- A340657 have a twice-balanced factorization.

Cf. A003963, A117409, A303975, A320656, A324518, A339846, A339890, A340608.

Sequence in context: A264857 A370645 A340596 * A303837 A286520 A320105

Adjacent sequences: A340651 A340652 A340653 * A340655 A340656 A340657

Keyword

nonn

Author

Gus Wiseman, Jan 15 2021

Status

approved