A340653 Number of balanced factorizations of n.

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 1, 0, 0, 1, 2, 1, 3, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 1, 3, 1, 2, 2, 0, 1, 2, 0, 2, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 0, 2, 1, 0, 3, 1, 2, 0, 3, 1, 3, 1, 0, 2, 2, 0, 3, 1, 2, 1, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 2, 0, 0, 0, 3, 1, 2, 2, 2, 1, 3, 1, 1, 3, 0, 1, 3, 1, 3, 0, 2, 1, 3, 0, 2, 2, 0, 0, 4

Offset

1,12

Comments

A factorization into factors > 1 is balanced if it is empty or its length is equal to its maximum Omega (A001222).

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

Example

The balanced factorizations for n = 120, 144, 192, 288, 432, 768:
  3*5*8    2*8*9    3*8*8      4*8*9      6*8*9      8*8*12
  2*2*30   3*6*8    4*6*8      6*6*8      2*8*27     2*2*8*24
  2*3*20   2*4*18   2*8*12     2*8*18     3*8*18     2*3*8*16
  2*5*12   2*6*12   4*4*12     3*8*12     4*4*27     2*4*4*24
           3*4*12   2*2*2*24   4*4*18     4*6*18     2*4*6*16
                    2*2*3*16   4*6*12     4*9*12     3*4*4*16
                               2*12*12    6*6*12     2*2*12*16
                               2*2*2*36   2*12*18    2*2*2*2*48
                               2*2*3*24   3*12*12    2*2*2*3*32
                               2*3*3*16   2*2*2*54
                                          2*2*3*36
                                          2*3*3*24
                                          3*3*3*16

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[PrimeOmega/@#]&]], {n, 100}]

Prog

(PARI) A340653(n, m=n, mbo=0, e=0) = if(1==n, mbo==e, sumdiv(n, d, if((d>1)&&(d<=m), A340653(n/d, d, max(mbo, bigomega(d)), 1+e)))); \\ Antti Karttunen, Oct 22 2023

Crossrefs

Positions of zeros are A001358.

Positions of nonzero terms are A100959.

The co-balanced version is A340596.

Taking maximum factor instead of maximum Omega gives A340599.

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.

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.

- A340600 counts unlabeled balanced multiset partitions.

- A340656 have no twice-balanced factorizations.

- A340657 have a twice-balanced factorization.

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

Sequence in context: A219840 A343220 A264893 * A334744 A136567 A336569

Adjacent sequences: A340650 A340651 A340652 * A340654 A340655 A340656

Keyword

nonn

Author

Gus Wiseman, Jan 15 2021

Extensions

Data section extended up to a(120) by Antti Karttunen, Oct 22 2023

Status

approved