A340853 Number of factorizations of n such that every factor is a multiple of the number of factors.

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

Offset: 1

Author

Gus Wiseman, Feb 04 2021

Keywords

• nonn

Comments

Also factorizations whose greatest common divisor is a multiple of the number of factors.

Examples

			The a(n) factorizations for n = 2, 4, 16, 48, 96, 144, 216, 240, 432:
  2   4     16    48     96     144     216      240     432
      2*2   2*8   6*8    2*48   2*72    4*54     4*60    6*72
            4*4   2*24   4*24   4*36    6*36     6*40    8*54
                  4*12   6*16   6*24    12*18    8*30    12*36
                         8*12   8*18    2*108    10*24   18*24
                                12*12   6*6*6    12*20   2*216
                                        3*3*24   2*120   4*108
                                        3*6*12           3*3*48
                                                         3*6*24
                                                         6*6*12
                                                         3*12*12
		

Crossrefs

Positions of 1's are A048103.

Positions of terms > 1 are A100716.

The version for partitions is A143773 (A316428).

The reciprocal for partitions is A340693 (A340606).

The version for strict partitions is A340830.

The reciprocal version is A340851.

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 factors, even-length case A340786.

A340831/A340832 counts factorizations with odd maximum/minimum.

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

Cf. A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340654, A340655, A340827.

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],n>1&&Divisible[GCD@@#,Length[#]]&]],{n,100}]