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A303709 Number of periodic factorizations of n using elements of A007916 (numbers that are not perfect powers). 8
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

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

Sequence in context: A074038 A204843 A204853 * A185778 A071164 A027345

Adjacent sequences:  A303706 A303707 A303708 * A303710 A303711 A303712

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 29 2018

EXTENSIONS

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

STATUS

approved

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Last modified November 21 06:00 EST 2019. Contains 329350 sequences. (Running on oeis4.)