A303709 Number of periodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Apr 29 2018

Keywords

• nonn

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).

Examples

			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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A281116, A303547, A303552, A303553, A303386, A303707, A303708, A303710.

Programs

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}]

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

Formula

a(n) <= A303553(n) <= A001055(n). - Antti Karttunen, Dec 06 2018

Extensions

Changed a(1) to 1 by Gus Wiseman, Dec 06 2018