A319058 Number of relatively prime aperiodic factorizations of n into factors > 1.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 4, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 4, 0, 2, 2, 1, 0, 5, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 8, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 9, 0, 1, 2, 2, 1, 4, 0, 5, 0, 1, 0, 8, 1, 1, 1

Offset

1,12

Comments

A factorization is relatively prime if the factors have no common divisor > 1, and aperiodic if the factors appear with relatively prime multiplicities.

Links

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

Example

The a(144) = 14 factorizations:
  (2*2*2*2*9), (2*2*2*3*6), (2*2*3*3*4),
  (2*2*3*12), (2*2*4*9), (2*3*3*8), (2*3*4*6),
  (2*3*24), (2*8*9), (3*3*16), (3*4*12), (3*6*8), (4*4*9),
  (9*16).
The 2 relatively prime but not aperiodic factorizations missing from this list are (2*2*2*2*3*3) and (3*3*4*4), while the 11 missing factorizations that are not relatively prime but are aperiodic are (2*2*2*18), (2*2*36), (2*4*18), (2*6*12), (2*72), (3*48), (4*6*6), (4*36), (6*24), (8*18), and (144), while the 2 missing factorizations that are neither relatively prime nor aperiodic are (2*2*6*6) and (12*12), for a total of A001055(144) = 29 factorizations.

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], And[GCD@@#==1, GCD@@Length/@Split[#]==1]&]], {n, 100}]

Crossrefs

Cf. A000740, A000837, A001055, A007916, A045778, A100953, A162247, A281116, A303386.

Sequence in context: A354911 A367098 A343660 * A281116 A335447 A089233

Adjacent sequences: A319055 A319056 A319057 * A319059 A319060 A319061

Keyword

nonn

Author

Gus Wiseman, Sep 09 2018

Status

approved