A340851 Number of factorizations of n such that every factor is a divisor of the number of factors.

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

Offset

1,64

Comments

Also factorizations whose number of factors is divisible by their least common multiple.

Links

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

Example

The a(n) factorizations for n = 8192, 46656, 73728:
  2*2*2*2*2*4*8*8          6*6*6*6*6*6              2*2*2*2*2*2*2*2*2*4*6*6
  2*2*2*2*4*4*4*8          2*2*2*2*2*2*3*3*3*3*3*3  2*2*2*2*2*2*2*2*3*4*4*6
  2*2*2*4*4*4*4*4                                   2*2*2*2*2*2*2*3*3*4*4*4
  2*2*2*2*2*2*2*2*2*2*2*4                           2*2*2*2*2*2*2*2*2*2*6*12
                                                    2*2*2*2*2*2*2*2*2*3*4*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], And@@IntegerQ/@(Length[#]/#)&]], {n, 100}]

Crossrefs

The version for partitions is A340693, with reciprocal version A143773.

Positions of nonzero terms are A340852.

The reciprocal version is A340853.

A320911 can be factored into squarefree semiprimes.

A340597 have an alt-balanced factorization.

A340656 lack a twice-balanced factorization, complement A340657.

- Factorizations -

A001055 counts factorizations, with strict case A045778.

A316439 counts factorizations by product and length.

A339846 counts factorizations of even length.

A339890 counts factorizations of odd length.

A340101 counts factorizations into odd factors, odd-length case A340102.

A340653 counts balanced factorizations.

A340785 counts factorizations into even numbers, even-length case A340786.

A340831/A340832 count factorizations with odd maximum/minimum.

A340854 cannot be factored with odd least factor, complement A340855.

Cf. A067538, A074761, A168659, A301987, A327517, A340596, A340599, A340654, A340655, A340827, A340830.

Sequence in context: A083919 A063665 A276306 * A130779 A130706 A000038

Adjacent sequences: A340848 A340849 A340850 * A340852 A340853 A340854

Keyword

nonn

Author

Gus Wiseman, Feb 04 2021

Status

approved