A157612 -id:A157612 - cp's OEIS frontend

A076716 Number of distinct factorizations of n! with all factors > 1.

1, 1, 2, 7, 21, 98, 392, 2116, 11830, 70520, 425240, 2787810, 19530213, 144890639, 1149978830, 8558078111, 76417516719, 618437486332, 6087770992601, 54574732902278, 525656554130914, 5290117056157616, 61626071051832409, 555057889968635744, 5809502058957961682

Offset: 1

Donald S. McDonald, Oct 27 2002

nonn

			a(3) = 2 because 3! = 6 = 2*3 has just 2 factorizations.
a(4) = 7 because 4! = 24 = 2*12 = 2*2*6 = 2*2*2*3 = 2*3*4 = 3*8 = 4*6 has 7 factorizations.

Cf. A001055, A157612, A162247, A318284.

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(n!$2):
seq(a(n), n=1..12);  # Alois P. Heinz, May 25 2013

c[1, r_] := c[1, r]=1; c[n_, r_] := c[n, r]=Module[{ds, i}, ds=Select[Divisors[n], 1<#<=r&]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n!, n! ]; a/@Range[16] (* c[n, r] is the number of factorizations of n with factors <= r. - Dean Hickerson, Oct 29 2002 *)

\\ See A318284 for count.
a(n)={if(n<=1, 1, count(factor(n!)[,2]))} \\ Andrew Howroyd, Feb 01 2020

a(n) = A001055(n!).

Edited by Robert G. Wilson v, Oct 29 2002
4 more terms from Ryan Propper, May 20 2007
a(20)-a(25) from Andrew Howroyd, Feb 01 2020

A321468 Number of factorizations of n! into factors > 1 that can be obtained by taking the multiset union of a choice of factorizations of each positive integer from 2 to n into factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 10, 20, 40, 40, 116, 116, 232, 464, 1440, 1440, 4192, 4192, 11640, 23280, 46560, 46560, 157376

Offset: 0

Gus Wiseman, Nov 11 2018

a(n) is the number of factorizations finer than (2*3*...*n) in the poset of factorizations of n! into factors > 1, ordered by refinement.

			The a(2) = 1 through a(8) = 10 factorizations:
2  2*3  2*3*4    2*3*4*5    2*3*4*5*6      2*3*4*5*6*7      2*3*4*5*6*7*8
        2*2*2*3  2*2*2*3*5  2*2*2*3*5*6    2*2*2*3*5*6*7    2*2*2*3*5*6*7*8
                            2*2*3*3*4*5    2*2*3*3*4*5*7    2*2*3*3*4*5*7*8
                            2*2*2*2*3*3*5  2*2*2*2*3*3*5*7  2*2*3*4*4*5*6*7
                                                            2*2*2*2*3*3*5*7*8
                                                            2*2*2*2*3*4*5*6*7
                                                            2*2*2*3*3*4*4*5*7
                                                            2*2*2*2*2*2*3*5*6*7
                                                            2*2*2*2*2*3*3*4*5*7
                                                            2*2*2*2*2*2*2*3*3*5*7
For example, 2*2*2*2*2*2*3*5*6*7 = (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2), so (2*2*2*2*2*2*3*5*6*7) is counted under a(8).

Dominated by A321514.

Cf. A001055, A066723, A076716, A157612, A242422, A265947, A300383, A317144, A317145, A317534, A321467, A321470, A321471, A321472.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Union[Sort/@Join@@@Tuples[facs/@Range[2,n]]]],{n,10}]

A321467 Number of factorizations of n! into factors > 1 that can be obtained by taking the block-products of some set partition of {2,3,...,n}.

Original entry on oeis.org

1, 1, 1, 2, 5, 15, 47, 183, 719, 3329, 14990, 83798, 393864, 2518898

Offset: 0

Gus Wiseman, Nov 11 2018

a(n) is the number of factorizations coarser than (2*3*...*n) in the poset of factorizations of n! into factors > 1, ordered by refinement.

			The a(1) = 1 through a(5) = 15 factorizations:
  ()  (2)  (6)    (24)     (120)
           (2*3)  (3*8)    (2*60)
                  (4*6)    (3*40)
                  (2*12)   (4*30)
                  (2*3*4)  (5*24)
                           (6*20)
                           (8*15)
                           (10*12)
                           (3*5*8)
                           (4*5*6)
                           (2*3*20)
                           (2*4*15)
                           (2*5*12)
                           (3*4*10)
                           (2*3*4*5)
For example, 10*12 = (2*5)*(3*4), so (10*12) is counted under a(5).

Dominated by A000110.

Cf. A001055, A066723, A157612, A242422, A265947, A317141, A317144, A317145, A317534, A321468, A321470, A321471, A321472, A321514.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Union[Sort/@Apply[Times,sps[Range[2,n]],{2}]]],{n,10}]

A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 12, 24, 48, 48, 192, 192, 384, 768, 3840, 3840, 15360, 15360, 61440, 122880, 245760, 245760, 1720320, 3440640, 6881280, 20643840, 82575360, 82575360, 412876800, 412876800, 2890137600, 5780275200, 11560550400, 23121100800, 208089907200

Offset: 1

Gus Wiseman, Nov 11 2018

nonn

			The a(8) = 12 ways to choose a factorization of each integer from 2 to 8:
  (2)*(3)*(4)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*2*2)

Dominates A321468.

Cf. A001055, A050336, A066723, A076716, A157612, A281113, A300383, A317144, A317145, A321467, A321470, A321471, A321472.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Array[Length[facs[#]]&,n,1,Times],{n,30}]

a(n) = Product_{k = 1..n} A001055(k).

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

A103774 Number of ways to write n! as product of squarefree numbers.

Original entry on oeis.org

1, 1, 2, 2, 6, 10, 42, 42, 82, 204, 1196, 1556, 10324, 34668, 104948, 104964, 873540, 1309396, 11855027, 25238220, 91193575, 453628255, 5002616219, 5902762219, 21142729523, 122981607092, 189706055368, 547296181656, 7291700021313, 14330422534833, 202498591157970

Offset: 1

Reinhard Zumkeller, Feb 15 2005

nonn

a(n) = A050320(A000142(n)).
From Gus Wiseman, Aug 20 2020: (Start)
Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of n!. For example, The a(2) = 1 through a(6) = 10 set multipartitions are:
  {1}  {12}    {1}{1}{12}    {1}{1}{123}      {1}{1}{12}{123}
       {1}{2}  {1}{1}{1}{2}  {1}{12}{13}      {1}{12}{12}{13}
                             {1}{1}{1}{23}    {1}{1}{1}{12}{23}
                             {1}{1}{2}{13}    {1}{1}{1}{2}{123}
                             {1}{1}{3}{12}    {1}{1}{2}{12}{13}
                             {1}{1}{1}{2}{3}  {1}{1}{3}{12}{12}
                                              {1}{1}{1}{1}{2}{23}
                                              {1}{1}{1}{2}{2}{13}
                                              {1}{1}{1}{2}{3}{12}
                                              {1}{1}{1}{1}{2}{2}{3}
(End)

			n=5, 5! = 1*2*3*4*5 = 120 = 2 * 2 * 2 * 3 * 5: a(5)=#{2*2*2*3*5,2*2*2*15,2*2*6*5,2*2*30,2*2*3*10,2*6*10}=6.

Index entries for sequences related to factorial numbers.

A103775 is the strict case.
A157612 is the case of superprimorials.
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.
A076716 counts factorizations of factorials.
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
A157612 counts strict factorizations of factorials.
Cf. A000110, A005117, A008480, A124010, A318360.
Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.

sub[w_, e_] := Block[{v=w}, v[[e]]--; v]; ric[w_, k_] := 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}]]]; a[n_] := ric[ Sort[ Last /@ FactorInteger[n!]], 1]; Array[a, 22] (* Giovanni Resta, Sep 30 2019 *)

a(17)-a(18) from Amiram Eldar, Sep 30 2019

a(19)-a(31) from Giovanni Resta, Sep 30 2019

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)).

A157672 Number of unordered factorizations of n! into two distinct proper factors.

Original entry on oeis.org

0, 1, 3, 7, 14, 29, 47, 79, 134, 269, 395, 791, 1295, 2015, 2687, 5375, 7343, 14687, 20519, 30399, 47999, 95999, 121439, 170015, 266111, 338687, 458639, 917279, 1166399, 2332799, 2764799, 3932159, 6082559, 8211455, 9797759, 19595519

Offset: 2

Jaume Oliver Lafont, Mar 04 2009, Mar 05 2009

nonn

Cf. A157612.

Table[Times@@(Last/@FactorInteger[ n! ]+1)/2-1,{n,2,40}] (* Ray Chandler, Mar 07 2009 *)

for(k=2,40,print1(numdiv(k!)/2-1,", "))

from math import prod
from collections import Counter
from sympy import factorint
def A157672(n): return prod(e+1 for e in sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values())//2-1 # Chai Wah Wu, Jun 25 2022

For n > 1, a(n) = A027423(n)/2 - 1. - Ray Chandler, Mar 07 2009

Extended by Ray Chandler, Mar 07 2009

PARI program rewritten Jaume Oliver Lafont, Mar 09 2009

A157836 Triangle read by rows where T(n,k) is the number of factorizations of (n+1)! into k distinct factors.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 14, 28, 13, 1, 1, 29, 103, 95, 24, 1, 1, 47, 273, 448, 249, 41, 1, 1, 79, 725, 1897, 1837, 671, 74, 1, 1, 134, 1876, 7301, 10856, 6780, 1686, 127, 1, 1, 269, 5791, 31811, 65782, 59434, 24017, 3960, 197, 1, 1, 395, 12061, 92987, 272932, 362956, 232152, 69765, 8703, 323, 1

Offset: 1

Ray Chandler, Mar 07 2009

n-th row has n terms; first and last term in each row = 1.

			Triangle begins:
2! 1
3! 1 1
4! 1 3 1
5! 1 7 7 1
6! 1 14 28 13 1
7! 1 29 103 95 24 1
8! 1 47 273 448 249 41 1
9! 1 79 725 1897 1837 671 74 1
10! 1 134 1876 7301 10856 6780 1686 127 1
11! 1 269 5791 31811 65782 59434 24017 3960 197 1
12! 1 395 12061 92987 272932 362956 232152 69765 8703 323 1
...

Andrew Howroyd, Table of n, a(n) for n = 1..465 (first 30 rows)

A157612 gives row sums. A157672 gives 2nd column.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n, sig)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(r=EulerT(v)); prod(i=1, #sig, r[sig[i]])/prod(i=1, #v, i^v[i]*v[i]!)}
detail(sig)={my(m=vecsum(sig)+1,n=vecmax(sig), q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(y+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, sig), [1, n]); s*q[#q-j]*y^m)/(1+y))}
row(n)={if(n<=1, [], Vecrev(detail(factor(n!)[,2])))}
{ for(n=1, 10, print(row(n+1))) } \\ Andrew Howroyd, Feb 01 2020

cp's OEIS Frontend

A076716 Number of distinct factorizations of n! with all factors > 1.

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A321468 Number of factorizations of n! into factors > 1 that can be obtained by taking the multiset union of a choice of factorizations of each positive integer from 2 to n into factors > 1.

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A321467 Number of factorizations of n! into factors > 1 that can be obtained by taking the block-products of some set partition of {2,3,...,n}.

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Mathematica

A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

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Mathematica

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

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Mathematica

PARI

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Extensions

A103774 Number of ways to write n! as product of squarefree numbers.

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Mathematica

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

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Mathematica

Formula

A157672 Number of unordered factorizations of n! into two distinct proper factors.

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