A337072 -id:A337072 - 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

A337069 Number of strict factorizations of the superprimorial A006939(n).

Original entry on oeis.org

1, 1, 3, 34, 1591, 360144, 442349835, 3255845551937, 156795416820025934, 53452979022001011490033, 138542156296245533221812350867, 2914321438328993304235584538307144802, 528454951438415221505169213611461783474874149, 873544754831735539240447436467067438924478174290477803

Offset: 0

Gus Wiseman, Aug 15 2020

nonn

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

Also the number of strict multiset partitions of {1,2,2,3,3,3,...,n}, a multiset with i copies of i for i = 1..n.

			The a(3) = 34 factorizations:
  2*3*4*15  2*3*60   2*180  360
  2*3*5*12  2*4*45   3*120
  2*3*6*10  2*5*36   4*90
  2*4*5*9   2*6*30   5*72
  3*4*5*6   2*9*20   6*60
            2*10*18  8*45
            2*12*15  9*40
            3*4*30   10*36
            3*5*24   12*30
            3*6*20   15*24
            3*8*15   18*20
            3*10*12
            4*5*18
            4*6*15
            4*9*10
            5*6*12
            5*8*9

A022915 counts permutations of the same multiset.
A157612 is the version for factorials instead of superprimorials.
A317829 is the non-strict version.
A337072 is the non-strict version with squarefree factors.
A337073 is the case with squarefree factors.
A000217 counts prime factors (with multiplicity) of superprimorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A045778 counts strict factorizations.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A181818 lists products of superprimorials, with complement A336426.
A322583 counts factorizations into factorials.
Cf. A000142, A000178, A002110, A022559, A027423, A303279, A318286, A322583, A337070, A337071.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
stfa[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[stfa[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Length[stfa[chern[n]]],{n,0,3}]

\\ See A318286 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Sep 01 2020

a(n) = A045778(A006939(n)).

a(n) = A318286(A002110(n)). - Andrew Howroyd, Sep 01 2020

a(7)-a(13) from Andrew Howroyd, Sep 01 2020

A103775 Number of ways to write n! as product of distinct squarefree numbers.

Original entry on oeis.org

1, 1, 2, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Feb 15 2005

nonn

From Gus Wiseman, Aug 23 2020: (Start)
Also the number of set-systems (sets of sets) whose multiset union is the multiset of prime factors of n!. For example, the a(1) = 1 through a(7) = 3 set-systems (empty columns indicated by dots) are:
  0  {1}  {1,2}    .  {1},{1,2},{1,3}  .  {1},{1,2},{1,3},{1,2,4}
          {1},{2}                         {1},{1,2},{1,4},{1,2,3}
                                          {1},{2},{1,2},{1,3},{1,4}
(End)

			n=7, 7! = 1*2*3*4*5*6*7 = 5040 = 2*2*2*2*3*3*5*7: a(7) = #{2*3*6*10*14, 2*6*10*42, 2*6*14*30} = 3.

Index entries for sequences related to factorial numbers.

A103774 is the non-strict version.
A337073 is the version for superprimorials, with non-strict version A337072.
A001055 counts factorizations.
A045778 counts strict factorizations.
A048656 counts squarefree divisors of factorials.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A050342 counts set-systems by total sum.
A076716 counts factorizations of factorials.
A116539 counts set-systems covering an initial interval.
A157612 counts strict factorizations of factorials.
Cf. A000110, A005117, A008480, A089259, A116540, A124010, A318360.
Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.

yst[n_]:=yst[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[yst[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[yst[n!]],{n,15}] (* Gus Wiseman, Aug 21 2020 *)

a(n) = 0 for n > 7;

a(n) = A050326(A000142(n)).

A337073 Number of strict factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

Original entry on oeis.org

1, 1, 1, 2, 14, 422, 59433, 43181280, 178025660042, 4550598470020490, 782250333882971717562, 974196106965358319940100513, 9412280190038329162111356578977100, 751537739224674099813783040471383322758327

Offset: 0

Gus Wiseman, Aug 15 2020

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1). It has n! divisors.

Also the number of strict set multipartitions (sets of sets) of the multiset of prime factors of the superprimorial A006939(n).

			The a(1) = 1 through a(3) = 10 factorizations:
    2  2*6  2*6*30    2*6*30*210
            2*3*6*10  6*10*30*42
                      2*3*6*30*70
                      2*5*6*30*42
                      2*3*10*30*42
                      2*3*6*10*210
                      2*6*10*15*42
                      2*6*10*21*30
                      2*6*14*15*30
                      3*6*10*14*30
                      2*3*5*6*10*42
                      2*3*5*6*14*30
                      2*3*6*7*10*30
                      2*3*6*10*14*15
The a(1) = 1 through a(3) = 14 set multipartitions:
    {1}  {1}{12}  {1}{12}{123}    {1}{12}{123}{1234}
                  {1}{2}{12}{13}  {12}{13}{123}{124}
                                  {1}{12}{13}{23}{124}
                                  {1}{12}{13}{24}{123}
                                  {1}{12}{14}{23}{123}
                                  {1}{2}{12}{123}{134}
                                  {1}{2}{12}{13}{1234}
                                  {1}{2}{13}{123}{124}
                                  {1}{3}{12}{123}{124}
                                  {2}{12}{13}{14}{123}
                                  {1}{2}{12}{13}{14}{23}
                                  {1}{2}{12}{4}{13}{123}
                                  {1}{2}{3}{12}{13}{124}
                                  {1}{2}{3}{12}{14}{123}

A000142 counts divisors of superprimorials.
A022915 counts permutations of the same multiset.
A103775 is the version for factorials instead of superprimorials.
A337072 is the non-strict version.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A045778 counts strict factorizations.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A050342 counts strict set multipartitions of integer partitions.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A283877 counts non-isomorphic strict set multipartitions.
A317829 counts factorizations of superprimorials.
A337069 counts strict factorizations of superprimorials.
Cf. A000178, A002110, A022559, A103774, A124010, A181818, A318361, A336417, A337070.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
ystfac[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[ystfac[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[ystfac[chern[n]]],{n,0,4}]

\\ See A318361 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Sep 01 2020

a(n) = A050326(A006939(n)).

a(n) = A318361(A002110(n)). - Andrew Howroyd, Sep 01 2020

a(7)-a(13) from Andrew Howroyd, Sep 01 2020

cp's OEIS Frontend

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

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A337069 Number of strict factorizations of the superprimorial A006939(n).

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A103775 Number of ways to write n! as product of distinct squarefree numbers.

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A337073 Number of strict factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

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