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A050320 Number of ways n is a product of squarefree numbers > 1. 45
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
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)
LINKS
Kevin Broughan, Quadrafree factorisatio numerorum, Rocky Mountain J. Math. 44 (3) (2014) 791-807.
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
EXAMPLE
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
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 *)
PROG
(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
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.
Sequence in context: A339887 A259936 A354870 * A333175 A294893 A336570
KEYWORD
nonn,easy,nice
AUTHOR
Christian G. Bower, Sep 15 1999
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)