A050320 Number of ways n is a product of squarefree numbers > 1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 3, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 6, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 5, 1

Offset: 1

Author

Christian G. Bower, Sep 15 1999

Keywords

• nonn
• easy
• nice

Comments

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

Broughan shows (Theorem 8) that the average value of a(n) is k exp(2*sqrt(log n)/sqrt(zeta(2)))/log(n)^(3/4) where k is about 0.18504. - Charles R Greathouse IV, May 21 2013

From Gus Wiseman, Aug 20 2020: (Start)

Also the number of set multipartitions (multisets of sets) of the multiset of prime indices of n. For example, the a(n) set multipartitions for n = 2, 6, 36, 60, 360 are:

{1} {12} {12}{12} {1}{123} {1}{12}{123}

{1}{2} {1}{2}{12} {12}{13} {12}{12}{13}

{1}{1}{2}{2} {1}{1}{23} {1}{1}{12}{23}

{1}{2}{13} {1}{1}{2}{123}

{1}{3}{12} {1}{2}{12}{13}

{1}{1}{2}{3} {1}{3}{12}{12}

{1}{1}{1}{2}{23}

{1}{1}{2}{2}{13}

{1}{1}{2}{3}{12}

{1}{1}{1}{2}{2}{3}

(End)

Examples

			For n = 36 we have three choices as 36 = 2*2*3*3 = 6*6 = 2*3*6 (but no factorizations with factors 4, 9, 12, 18 or 36 are allowed), thus a(36) = 3. - _Antti Karttunen_, Oct 21 2017
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• Kevin Broughan, Quadrafree factorisatio numerorum, Rocky Mountain J. Math. 44 (3) (2014) 791-807.
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A001055, A005117, A050325. a(p^k)=1. a(A002110)=A000110.

a(n!)=A103774(n).

Cf. A206778.

Differs from A259936 for the first time at n=36.

A050326 is the strict case.

A045778 counts strict factorizations.

A089259 counts set multipartitions of integer partitions.

A116540 counts normal set multipartitions.

Cf. A008480, A124010, A317829, A318360.

Programs

Haskell

a050320 n = h n $ tail $ a206778_row n where
   h 1 _          = 1
   h _ []         = 0
   h m fs'@(f:fs) =
     if f > m then 0 else if r > 0 then h m fs else h m' fs' + h m fs
     where (m', r) = divMod m f
-- Reinhard Zumkeller, Dec 16 2013

Mathematica

sub[w_, e_] := Block[{v = w}, v[[e]]--; v]; ric[w_, k_] := If[Max[w] == 0, 1, Block[{e, s, p = Flatten@Position[Sign@w, 1]}, s = Select[Prepend[#, First@p] & /@ Subsets[Rest@p], Total[1/2^#] <= k &]; Sum[ric[sub[w, e], Total[1/2^e]], {e, s}]]]; sig[w_] := sig[w] = ric[w, 1];  a[n_] := sig@ Sort[Last /@ FactorInteger[n]]; Array[a, 103] (* Giovanni Resta, May 21 2013 *)
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]]
Table[Length[sqfacs[n]],{n,100}] (* Gus Wiseman, Aug 20 2020 *)

Formula

Dirichlet g.f.: Product_{n is squarefree and > 1} (1/(1-1/n^s)).

a(n) = A050325(A101296(n)). - R. J. Mathar, May 26 2017

a(n!) = A103774(n); a(A006939(n)) = A337072(n). - Gus Wiseman, Aug 20 2020