OFFSET
1,36
COMMENTS
A periodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities have a common divisor greater than 1. Note that a factorization of a number that is not a perfect power (A007916) is always aperiodic (A303386), so the indices of nonzero entries of this sequence all lie at perfect powers (A001597).
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
FORMULA
EXAMPLE
The a(900) = 5 periodic factorizations are (2*2*3*3*5*5), (2*2*15*15), (3*3*10*10), (5*5*6*6), (30*30).
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, 200}]
PROG
(PARI)
gcd_of_multiplicities(lista) = { my(u=length(lista)); if(u<2, u, my(g=0, pe = lista[1], j=1); for(i=2, u, if(lista[i]==pe, j++, g = gcd(j, g); j=1; pe = lista[i])); gcd(g, j)); }; \\ the supplied lista (newfacs) should be monotonic
A303709(n, m=n, facs=List([])) = if(1==n, (1!=gcd_of_multiplicities(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m)&&!ispower(d), newfacs = List(facs); listput(newfacs, d); s += A303709(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Dec 06 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 29 2018
EXTENSIONS
Changed a(1) to 1 by Gus Wiseman, Dec 06 2018
STATUS
approved