A050325 -id:A050325 - cp's OEIS frontend

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

Christian G. Bower, Sep 15 1999

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)

			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

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

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.

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

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 *)

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

A051282 2-adic valuation of A025487: largest k such that 2^k divides A025487(n), where A025487 gives products of primorials.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 3, 1, 5, 2, 4, 2, 6, 3, 5, 3, 7, 4, 2, 6, 1, 3, 4, 8, 5, 3, 7, 2, 4, 5, 9, 6, 4, 8, 3, 5, 2, 6, 10, 3, 7, 2, 4, 5, 9, 4, 6, 3, 7, 11, 4, 8, 1, 3, 5, 6, 10, 5, 7, 4, 8, 12, 5, 9, 2, 4, 6, 3, 7, 11, 2, 4, 6, 8, 5, 3, 9, 5, 13, 6, 10, 3, 5, 7, 4, 8, 12, 3, 5, 7, 9, 2, 6, 4, 10, 6, 14, 7, 11, 4, 6, 8, 5, 9, 13, 4, 6, 8, 3, 10, 3, 7, 1, 5, 11, 7, 4

Offset: 1

Alford Arnold

a(n) can be used for resorting A025487 and sequences indexed by A025487, e.g., A050322, A050323, A050324 and A050325.

a(n) is the number of primorial numbers (A002110) larger than 1 in the representation of A025487(n) as a product of primorial numbers. - Amiram Eldar, Jun 03 2023

			a(8) = 3 because A025487(8) = 24 and 2^3 divides 24.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001055, A002033, A007814, A025487, A045778, A050320, A051903.

Cf. A050322, A050323, A050324, A050325.

a051282 = a007814 . a025487  -- Reinhard Zumkeller, Apr 06 2013

max = 40000; A025487 = {1}; lpe = {}; Do[ pe = Sort[ FactorInteger[n][[All, 2]]]; If[FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[A025487, n]], {n, 2, max}]; a[n_] := FactorInteger[ A025487[[n]] ][[1, 2]]; a[1] = 0; Table[a[n], {n, 1, Length[A025487]}] (* Jean-François Alcover, Jun 14 2012, after Robert G. Wilson v *)

isA025487(n)=my(k=valuation(n, 2), t); n>>=k; forprime(p=3, default(primelimit), t=valuation(n, p); if(t>k, return(0), k=t); if(k, n/=p^k, return(n==1)))
[valuation(n,2) | n <- [1..1000], isA025487(n)]
\\ Or, for older versions:
apply(n->valuation(n,2), select(isA025487, [1..1000])) \\ Charles R Greathouse IV, Nov 07 2014

a(n) = A007814(A025487(n)) = A051903(A025487(n)). - Matthew Vandermast, Jul 03 2012

More terms from Naohiro Nomoto, Mar 11 2001

cp's OEIS Frontend

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

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