|
|
A303708
|
|
Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).
|
|
12
|
|
|
0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 0, 2, 0, 3, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 0, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 0, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 0, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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.
|
|
LINKS
|
|
|
FORMULA
|
|
|
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
|
Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A100953, A275870, A281116, A301700, A303386, A303431, A303546, A303551, A303707, A303709, A303710.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|