OFFSET
1,6
COMMENTS
An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.
The positions of zeros in this sequence are the prime powers A000961.
EXAMPLE
The a(144) = 8 aperiodic factorizations are (2*2*2*3*6), (2*2*2*18), (2*2*3*12), (2*3*24), (2*6*12), (2*72), (3*48) and (6*24). Missing from this list are (12*12), (2*2*6*6) and (2*2*2*2*3*3).
MATHEMATICA
radQ[n_]:=Or[n===1, GCD@@FactorInteger[n][[All, 2]]===1];
facsr[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsr[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], radQ]}]];
Table[Length[Select[facsr[n], GCD@@Length/@Split[#]===1&]], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 29 2018
STATUS
approved