A303386 Number of aperiodic factorizations of n > 1.

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 2, 4, 1, 5, 1, 6, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 7, 1, 5, 1, 7, 5

Offset

2,5

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.

Links

Antti Karttunen, Table of n, a(n) for n = 2..65537

Formula

a(n) = Sum_{d|A052409(n)} mu(d) * A001055(n^(1/d)), where mu = A008683.

Example

The a(36) = 7 aperiodic factorizations are (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), and (36). Missing from this list are (2*2*3*3) and (6*6).

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], GCD@@Length/@Split[#]===1&]], {n, 2, 100}]

Prog

(PARI)
A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A303386(n) = if(1==n, n, my(r); sumdiv(A052409(n), d, ispower(n, d, &r); moebius(d)*A001055(r))); \\ Antti Karttunen, Sep 25 2018

Crossrefs

Cf. A000740, a(2^n) = A000837(n), A001055, A007716, A007916, A045778, A052409, A052410, A100953, A162247, A275024, A275870, A281113, A281116, A301700, A302242, A303431.

Sequence in context: A335434 A348379 A264440 * A295636 A050334 A342084

Adjacent sequences: A303383 A303384 A303385 * A303387 A303388 A303389

Keyword

nonn

Author

Gus Wiseman, Apr 23 2018

Extensions

More terms from Antti Karttunen, Sep 25 2018

Status

approved