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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 09 2018
STATUS
approved