A076716 -id:A076716 - cp's OEIS frontend

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

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)).

A325510 Number of non-isomorphic multiset partitions of the multiset of prime indices of n!.

Original entry on oeis.org

1, 1, 1, 2, 7, 16, 98, 269, 1397, 7582, 70520, 259906, 1677259, 5229112, 44726100, 666355170, 4917007185, 18459879921

Offset: 0

Gus Wiseman, May 08 2019

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 16 multiset partitions:
  {{1}}  {{12}}    {{1222}}        {{12333}}
         {{1}{2}}  {{1}{222}}      {{1}{2333}}
                   {{12}{22}}      {{12}{333}}
                   {{2}{122}}      {{13}{233}}
                   {{1}{2}{22}}    {{3}{1233}}
                   {{2}{2}{12}}    {{33}{123}}
                   {{1}{2}{2}{2}}  {{1}{2}{333}}
                                   {{1}{23}{33}}
                                   {{1}{3}{233}}
                                   {{3}{12}{33}}
                                   {{3}{13}{23}}
                                   {{3}{3}{123}}
                                   {{1}{1}{1}{23}}
                                   {{1}{2}{3}{33}}
                                   {{1}{3}{3}{23}}
                                   {{1}{2}{3}{3}{3}}

Cf. A000142, A001055, A007716, A011371, A022559, A076716, A115627, A317791, A318285, A322583, A325272, A325276, A325508, A325509, A325511.

\\ Requires C(sig) from A318285.
a(n)={if(n<2, 1, my(f=factor(n!)[,2], sig=vector(vecmax(f))); for(i=1, #f, sig[f[i]]++); C(sig))} \\ Andrew Howroyd, Jan 17 2023

a(n) = A317791(n!).

a(n) = A318285(A181819(n!)) = A318285(A325508(n)). - Andrew Howroyd, Jan 17 2023

a(9)-a(17) from Andrew Howroyd, Jan 17 2023

A336940 Number of odd divisors of n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 12, 12, 20, 30, 60, 72, 144, 216, 336, 336, 672, 864, 1728, 2160, 3200, 4800, 9600, 10560, 14784, 22176, 28224, 35280, 70560, 86400, 172800, 172800, 245760, 368640, 497664, 559872, 1119744, 1679616, 2363904, 2626560, 5253120, 6451200, 12902400, 16128000

Offset: 0

Gus Wiseman, Aug 23 2020

nonn

			The a(1) = 1 through a(8) = 12 divisors:
  1  1  1  1  1   1   1    1
        3  3  3   3   3    3
              5   5   5    5
              15  9   7    7
                  15  9    9
                  45  15   15
                      21   21
                      35   35
                      45   45
                      63   63
                      105  105
                      315  315

Seiichi Manyama, Table of n, a(n) for n = 0..10000

A049606 gives the maximum among these divisors, with quotient A060818.
A337257 is the even version.
A000265 gives the maximum odd divisor of n.
A001227 counts odd divisors.
A183063 counts even divisors.
Cf. A000005, A001013, A001055, A006939, A113474, A124010, A253249.
Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.

Table[Length[Select[Divisors[n!],OddQ]],{n,0,15}]

a(n) = sumdiv(n!, d, d%2); \\ Michel Marcus, Aug 24 2020

a(n) = numdiv(prod(k=1, n, k >> valuation(k, 2))); \\ Michel Marcus, Aug 27 2020

a(n) = A001227(n!).
a(n) = A000005(A049606(n)).
a(n) + A337257(n) = A027423(n) = A000005(n!).
From Seiichi Manyama, Aug 27 2020: (Start)
If p is odd prime, a(p) = 2 * a(p-1).
a(n) = A027423(n) / A113474(n) for n > 0. (End)

a(36)-a(44) from Seiichi Manyama, Aug 26 2020

A337106 Number of nontrivial divisors of n!.

Original entry on oeis.org

0, 0, 0, 2, 6, 14, 28, 58, 94, 158, 268, 538, 790, 1582, 2590, 4030, 5374, 10750, 14686, 29374, 41038, 60798, 95998, 191998, 242878, 340030, 532222, 677374, 917278, 1834558, 2332798, 4665598, 5529598, 7864318, 12165118, 16422910, 19595518, 39191038, 60466174

Offset: 0

Gus Wiseman, Aug 23 2020

nonn

A divisor of n is trivial if it is 1 or n.

			The a(3) = 2 through a(5) =14 nontrivial divisions:
  6/2  24/2   120/2
  6/3  24/3   120/3
       24/4   120/4
       24/6   120/5
       24/8   120/6
       24/12  120/8
              120/10
              120/12
              120/15
              120/20
              120/24
              120/30
              120/40
              120/60

A070824 counts nontrivial divisors.
A153823 counts proper divisors of n!.
A337107 has this sequence as column k = 3.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A337071 counts chains of divisors starting with n!.
A337105 counts chains of divisors from n! to 1.
Cf. A022559, A124010, A251683, A325617, A336941.

Table[Length[DeleteCases[Divisors[n!],1|n!]],{n,10}]

from sympy import factorial, divisor_count
def A337106(n):
    return 0 if n <= 1 else divisor_count(factorial(n))-2 # Chai Wah Wu, Aug 24 2020

a(n) = A000005(n!) - 2 for n > 1.

a(n) = A070824(n!).

a(0) from Chai Wah Wu, Aug 24 2020

A337257 Number of even divisors of n!.

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 24, 48, 84, 140, 240, 480, 720, 1440, 2376, 3696, 5040, 10080, 13824, 27648, 38880, 57600, 91200, 182400, 232320, 325248, 510048, 649152, 882000, 1764000, 2246400, 4492800, 5356800, 7618560, 11796480, 15925248

Offset: 0

Gus Wiseman, Aug 23 2020

nonn

			The a(2) = 1 through a(5) = 12 divisors:
  2  2  2   2
     6  4   4
        6   6
        8   8
        12  10
        24  12
            20
            24
            30
            40
            60
            120

Seiichi Manyama, Table of n, a(n) for n = 0..10000

A336940 is the odd version.
A000265 gives the maximum odd divisor of n.
A001227 counts odd divisors.
A183063 counts even divisors.
Cf. A000005, A001013, A001055, A006939, A049606, A124010, A253249.
Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.

Table[Length[Select[Divisors[n!],EvenQ]],{n,0,15}]

a(n) = sumdiv(n!, d, !(d%2)); \\ Michel Marcus, Aug 24 2020

a(n) = A183063(n!).

A336940(n) + a(n) = A027423(n) = A000005(n!).

A265626 Let S be the set of factorizations of n! where the largest factor is the largest prime <= n, and let f(s) be the least factor in the factorization s. Then a(n) = max f(S).

Original entry on oeis.org

2, 2, 2, 2, 3, 4, 3, 3, 4, 7, 6, 7, 7, 7, 7, 10, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37

Offset: 2

Charles R Greathouse IV, Dec 10 2015

nonn

			2! = 2
3! = 2 * 3
4! = 2^3 * 3
5! = 2 * 3 * 4 * 5
6! = 3^2 * 4^2 * 5
7! = 4 * 5 * 6^2 * 7
8! = 3 * 4^3 * 5 * 6 * 7
9! = 3 * 4^2 * 5 * 6^3 * 7
10! = 4^2 * 5^2 * 6^4 * 7
11! = 7 * 8^2 * 9^2 * 10^2 * 11
12! = 6^5 * 7 * 8 * 10^2 * 11
13! = 7 * 8^2 * 9^2 * 10^2 * 11 * 12 * 13
14! = 7^2 * 8 * 9 * 10^2 * 11 * 12^3 * 13
15! = 7^2 * 9 * 10^3 * 11 * 12^4 * 13
16! = 7^2 * 10^3 * 11 * 12^6 * 13
17! = 10 * 11 * 12^4 * 13 * 14^2 * 15^2 * 16 * 17

Cf. A001055, A007917, A007918, A076716, A177333.

f(n,mn,mx)=if(n%mn,return(0)); n/=mn; if(n==1, return(1)); for(k=mn,mx, if(f(n,k,mx), return(1))); 0
a(n)=if(n<6,return(2)); my(p=precprime(n),q=nextprime(p/2),N=n!); forprime(r=q+1,p-1, N/=r^valuation(N,r)); forstep(k=q,3,-1, if(f(N,k,p), return(k)))

a(n) > 5 for n > 10. a(n) < A007918(ceiling(A007917(n)/2)).

cp's OEIS Frontend

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

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A325510 Number of non-isomorphic multiset partitions of the multiset of prime indices of n!.

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A336940 Number of odd divisors of n!.

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A337106 Number of nontrivial divisors of n!.

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A337257 Number of even divisors of n!.

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A265626 Let S be the set of factorizations of n! where the largest factor is the largest prime <= n, and let f(s) be the least factor in the factorization s. Then a(n) = max f(S).

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