A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset

1

Comments

I call these alt-balanced factorizations. Balanced factorizations are A340653. - Gus Wiseman, Jan 20 2021

Links

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

Example

The alt-balanced factorizations for n = 192, 1728, 3456, 9216:
  3*4*4*4       2*2*2*6*6*6   2*2*4*6*6*6         4*4*4*4*6*6
  2*2*2*2*2*6   2*2*3*4*6*6   2*3*4*4*6*6         2*2*2*2*2*6*6*8
                2*3*3*4*4*6   3*3*4*4*4*6         2*2*2*2*3*3*8*8
                              2*2*2*2*3*3*3*8     2*2*2*2*3*4*6*8
                              2*2*2*2*2*2*2*3*9   2*2*2*3*3*4*4*8
                                                  2*2*2*2*2*2*2*8*9
                                                  2*2*2*2*2*2*4*4*9

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

Crossrefs

The co-balanced version is A340596.

Positions of nonzero terms are A340597.

The case of powers of two is A340611.

Taking maximum Omega instead of maximum factor gives 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.

- A340598 counts balanced set partitions.

- A340600 counts unlabeled balanced multiset partitions.

Cf. A006141, A117409, A320655, A320656, A324518, A339846, A339890, A340607.

Sequence in context: A193243 A281302 A369426 * A160753 A328981 A369070

Adjacent sequences: A340596 A340597 A340598 * A340600 A340601 A340602

Keyword

nonn

Author

Gus Wiseman, Jan 20 2021

Status

approved