A340655 Number of twice-balanced factorizations of n.

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 2, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 2, 0, 1, 0, 0, 2, 0, 2, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Jan 15 2021

Keywords

• nonn

Comments

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:

(1) the number of factors;

(2) the maximum image of A001222 over the factors;

(3) A001221(n).

Examples

			The twice-balanced factorizations for n = 12, 120, 360, 480, 900, 2520:
  2*6   3*5*8    5*8*9     2*8*30    2*6*75    2*2*7*90
  3*4   2*2*30   2*4*45    3*8*20    2*9*50    2*3*5*84
        2*3*20   2*6*30    4*4*30    3*4*75    2*3*7*60
        2*5*12   2*9*20    4*6*20    3*6*50    2*5*7*36
                 3*4*30    4*8*15    4*5*45    3*3*5*56
                 3*6*20    5*8*12    5*6*30    3*3*7*40
                 3*8*15    6*8*10    5*9*20    3*5*7*24
                 4*5*18    2*12*20   2*10*45   2*2*2*315
                 5*6*12    4*10*12   2*15*30   2*2*3*210
                 2*10*18             2*18*25   2*2*5*126
                 2*12*15             3*10*30   2*3*3*140
                 3*10*12             3*12*25
                                     3*15*20
                                     5*10*18
                                     5*12*15
		

Crossrefs

The co-balanced version is A340596.

The version for unlabeled multiset partitions is A340652.

The balanced version is A340653.

The cross-balanced version is A340654.

Positions of zeros are A340656.

Positions of nonzero terms are A340657.

A001055 counts factorizations.

A001221 counts distinct prime factors.

A001222 counts prime factors with multiplicity.

A045778 counts strict factorizations.

A303975 counts distinct prime factors in prime indices.

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 have an alt-balanced factorization.

- A340598 counts balanced set partitions.

- A340599 counts alt-balanced factorizations.

- A340600 counts unlabeled balanced multiset partitions.

Cf. A003963, A050336, A117409, A320655, A320656, A324518, A339846, A339890, A340607, A340608.

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[#]==PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,30}]