OFFSET
1,4
COMMENTS
Also factorizations whose greatest common divisor is a multiple of the number of factors.
EXAMPLE
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
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}]
CROSSREFS
Positions of 1's are A048103.
Positions of terms > 1 are A100716.
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.
- Factorizations -
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2021
STATUS
approved