A034878 -id:A034878 - cp's OEIS frontend

A001013 Jordan-Polya numbers: products of factorial numbers A000142.

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2304, 2592, 2880, 3072, 3456, 3840, 4096, 4320, 4608, 5040, 5184, 5760

Offset: 1

N. J. A. Sloane

Also, numbers of the form 1^d_1*2^d_2*3^d_3*...*k^d_k where k, d_1, ..., d_k are natural numbers satisfying d_1 >= d_2 >= d_3 >= ... >= d_k >= 1. - N. J. A. Sloane, Jun 14 2015
Possible orders of automorphism groups of trees.
Except for the numbers 2, 9 and 10 this sequence is conjectured to be the same as A034878.
Equivalently, (a(n)/6)*(6*x^2 - 6*x + (6*x-3)*a(n) + 2*a(n)^2 + 1) = N^2 has an integer solution. - Ralf Stephan, Dec 04 2004
Named after the French mathematician Camille Jordan (1838-1922) and the Hungarian mathematician George Pólya (1887-1985). - Amiram Eldar, May 22 2021
Possible numbers of transitive orientations of comparability graphs (Golumbic, 1977). - David Eppstein, Dec 29 2021

			864 = (3!)^2*4!.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B23, p. 123.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 987 terms from T. D. Noe)
Jean-Marie De Koninck, Nicolas Doyon, A. Arthur Bonkli Razafindrasoanaivolala and William Verreault, Bounds for the counting function of the Jordan-Pólya numbers, Archivum Mathematicum, Vol. 56, No. 3 (2020), pp. 141-152; also on arXiv, arXiv:2107.09114 [math.NT], 2021.
Martin Charles Golumbic, Comparability graphs and a new matroid, Journal of Combinatorial Theory, Series B, Vol. 22, No. 1 (1977), pp. 68-90.
Camille Jordan, Sur les assemblages de lignes, Journal für die reine und angewandte Mathematik, Vol. 70 (1869), pp. 185-190; alternative link.
Robert A. Melter, Autometrized unary algebras, J. Combinatorial Theory, Vol. 5, No. 1 (1968), pp. 21-29.
George Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica, Vol. 68 (1937), pp. 145-254; alternative link.
Eric Weisstein's World of Mathematics, Factorial Products.
Index entries for sequences related to factorial numbers
Index entries for sequences related to trees

Cf. A000142, A034878, A093373 (complement), A344438 (characteristic function).
Union of A344181 and A344179. Subsequence of A025487 (see also A064783).
See also A359636 and A359751.

import Data.Set (empty, fromList, deleteFindMin, union)
import qualified Data.Set as Set (null)
a001013 n = a001013_list !! (n-1)
a001013_list = 1 : h 0 empty [1] (drop 2 a000142_list) where
   h z s mcs xs'@(x:xs)
    | Set.null s || x < m = h z (union s (fromList $ map (* x) mcs)) mcs xs
    | m == z = h m s' mcs xs'
    | otherwise = m : h m (union s' (fromList (map (* m) $ init (m:mcs)))) (m:mcs) xs'
    where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Nov 13 2014

N:= 10000: # get all terms <= N
S:= {1}:
for k from 2 do
  kf:= k!;
  if kf > N then break fi;
  S := S union {seq(seq(kf^j * s, j = 1 .. floor(log[kf](N/s))),s=S)};
od:
S;   # if using Maple 11 or earlier, uncomment the next line:
# sort(convert(S,list));
# Robert Israel, Sep 09 2014

For[p=0; a=f=Table[n!, {n, 1, 8}], p=!=a, p=a; a=Select[Union@@Outer[Times, f, a], #<=8!&]]; a

list(lim,mx=lim)=if(lim<2, return([1])); my(v=[1],t=1); for(n=2,mx, t*=n; if(t>lim, break); v=concat(v,t*list(lim\t, t))); Set(v) \\ Charles R Greathouse IV, May 18 2015

def aupto(lim, mx=None):
    if lim < 2: return [1]
    v, t = [1], 1
    if mx == None: mx = lim
    for k in range(2, mx+1):
        t *= k
        if t > lim: break
        v += [t*rest for rest in aupto(lim//t, t)]
    return sorted(set(v))
print(aupto(5760)) # Michael S. Branicky, Jul 21 2021 after Charles R Greathouse IV

# uses[prod_hull from A246663]
prod_hull(factorial, 5760) # Peter Luschny, Sep 09 2014

More terms, formula from Christian G. Bower, Dec 15 1999

Edited by Dean Hickerson, Sep 17 2002

A075082 Numbers n such that n! is a product of distinct factorials k!l!m!*... with k, l, m, etc. < n.

Original entry on oeis.org

1, 6, 10, 12, 16, 24, 48, 120, 144, 240, 288, 720, 1440, 2880, 4320, 5040, 5760, 8640, 10080, 17280, 30240, 34560, 40320, 60480, 80640, 86400, 103680, 120960, 172800, 207360, 241920, 362880, 483840, 518400, 604800, 725760, 967680, 1036800

Offset: 1

Amarnath Murthy, Sep 11 2002

nonn

r! is a member for r>2, since (r!)! = (r!)*(r!-1)!.
Subsequence of A034878 (all n such that n! is a product of smaller factorials). It is conjectured that A034878 and A001013 (Jordan-Polya numbers = products of factorials) are the same sequence (except for the numbers 2, 9 and 10). If this is true, then obviously A075082 (without the number 10) is also a subsequence of A001013. On the other hand, this special case of the conjecture might be easier to prove. (a(n)!)^2 is a member of A058295 (products of distinct factorials); for example, (6!)^2 = 6!*5!*3!. - Jonathan Sondow, Dec 21 2004
May be the same as A058295 except for 2, 10 and 16. - Jud McCranie, Jun 13 2005
By using similar logic, r!s!t! is a member for at least two, all distinct r,s,t,... > 1. - Robert G. Wilson v, Jan 27 2006
Except for 1, 10 & 16, all the members are of the form immediately above. - Robert G. Wilson v, Jan 27 2006
Except for 10 and 16, all members, n, have as the greatest factorial in is product representation of n, n-1. - Robert G. Wilson v, Jan 27 2006
Theorem, for n to be a member of A075082, then the largest distinct factorial, m!, less than n! must not be less than the greatest prime less than n. - Robert G. Wilson v, Jan 27 2006

			1! = 0!, 6! = 5!*3!, 10! = 7!*6!, 12! = 11!*3!*2!, 16! = 14!*5!*2!,
24! = 23!*4!, 48! = 47!*4!*2!, 120! = 119!*5!, 144! = 143! *4!*3!,
240! = 239!*5!*2!, 288! = 287!*4!*3!*2!, 720! = 719!*6!,
1440! = 1439!6!*2!, etc.

R. K. Guy, Unsolved Problems in Number Theory, B23.

Index entries for sequences related to factorial numbers.

Cf. A034878, A001013, A058295.

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) s = Sort[ Table[ Times @@ Factorial /@ UnrankSubset[n, Table [i, {i, 2, 12}]], {n, 2047}]]; f[n_] := Block[{k = Prime[ PrimePi [n]]}, While[k < n && Position[s, Product[i, {i, k + 1, n}]] == {}, k+ + ]; If[k == n, 0, k]]; Do[a = f[n]; If[a != 0, Print[{n, a}]], {n, 3, 1210000}] (* Robert G. Wilson v, Jun 20 2005 *)

Corrected and extended by Jud McCranie, Sep 13 2002
More terms from Jud McCranie, Jun 13 2005
a(25)-a(39) proposed by Robert G. Wilson v, Jun 18 2005, confirmed by David Wasserman, Dec 30 2005

A003135 n! is a nontrivial product of factorials. It is conjectured that the list is complete.

Original entry on oeis.org

9, 10, 16

Offset: 1

N. J. A. Sloane

A "nontrivial" solution is one in which the largest x! in the product of a(n)! is such that x < a(n)-1. There are no other terms < 10^5. - Jud McCranie, Jun 15 2005

			9! = 2! * 3! * 3! * 7! and 7 < 9-1, so 9 is in the sequence.
10! = 6! * 7! or 10! = 3! * 5! * 7! and 7 < 10-1, so 10 is in the sequence.
16! = 2! * 5! * 14! and 14 < 16-1, so 16 is in the sequence.

R. K. Guy, "Unsolved Problems in Number Theory", section B23.

P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 25-44.
Index entries for sequences related to factorial numbers

Cf. A034878, A001013, A058295, A075082, A109095, A109096, A109097, A109098.

A109095 Numbers N such that N! is the product of exactly two smaller factorials (larger than 1).

Original entry on oeis.org

6, 10, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000, 25852016738884976640000

Offset: 1

Jud McCranie, Jun 19 2005

nonn

N = x! is considered to be a trivial solution because then N! = N*(N-1)! = x!*(N-1)!. Therefore every factorial appears in this sequence.
All terms except a(2) = 10 appear to be trivial solutions. (From Erdős's paper this is known as Surányi's conjecture.)
Habsieger established that the least nontrivial solution must have N > 10^3000. - M. F. Hasler, Jan 19 2023

			10! = 6! * 7!, so 10 is in the sequence.

Richard K. Guy, Unsolved Problems in Number Theory, B23 Equal products of factorials, Springer, Third Edition, 2004, p. 123.
Laurent Habsieger, Explicit bounds for the Diophantine equation A!B! = C!, Fibonacci Quarterly (2019), 57, 1.

Paul Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 25-44.
Laurent Habsieger, Explicit Bounds For The Diophantine Equation A!B! = C!, arXiv:1903.08370 [math.NT], 2019.

Cf. A000142, A034878, A001013, A003135, A058295, A075082, A109096, A109097.

is_A109095(n) = my(m=1, f=n!); while(n-->m, while(n!2));
select(is_A109095, [0..777]) \\ M. F. Hasler, Jan 19 2023

Definition corrected by Jon E. Schoenfield, Jul 02 2010

More terms from M. F. Hasler, Jan 19 2023

A109096 Numbers n such that n! is the product of exactly three smaller factorials.

Original entry on oeis.org

4, 10, 12, 16, 36, 48, 144, 240, 576, 720, 1440, 2880, 4320, 10080, 14400, 17280, 30240, 80640, 86400

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			144! = 3! * 4! * 143!, so 144 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109097.

Definition corrected by Jon E. Schoenfield, Jul 02 2010

A109097 Numbers n such that n! is the product of exactly four smaller factorials.

Original entry on oeis.org

8, 9, 24, 72, 96, 216, 288, 480, 864, 1152, 1440, 2880, 3456, 4320, 5760, 8640, 13824, 17280, 20160, 25920, 28800, 34560, 60480, 69120, 86400

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			86400! = 3! * 5! * 5! * 86399!, so 86400 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096.

Definition corrected by Jon E. Schoenfield, Jul 02 2010

A325509 Number of factorizations of n! into factorial numbers > 1.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 08 2019

nonn

			n = 10:
  (6*120*5040)
  (720*5040)
  (3628800)
n = 16:
  (2*2*2*2*1307674368000)
  (2*120*87178291200)
  (20922789888000)
n = 24:
  (2*2*6*25852016738884976640000)
  (24*25852016738884976640000)
  (620448401733239439360000)

Cf. A000142, A011371, A022559, A034876, A034878, A076934, A109100, A109101, A109102, A115627, A135291, A322583, A325272, A325273, A325276, A325508, A325510, A325511.

facs[n_,u_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d,u],Min@@#>=d&]],{d,Intersection[u,Rest[Divisors[n]]]}]];
Table[Length[facs[n!,Rest[Array[#!&,n]]]],{n,15}]

a(n) = 1 + A034876(n).

More terms from Alois P. Heinz, May 08 2019

A034876 Number of ways to write n! as a product of smaller factorials each greater than 1.

Original entry on oeis.org

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

Offset: 1

Erich Friedman

By definition, a(n) > 0 if and only if n is a member of A034878. If n > 2, then a(n!) > max(a(n), a(n!-1)), as (n!)! = n!*(n!-1)!. Similarly, a(A001013(n)) > 0 for n > 2. Clearly a(n)=0 if n is a prime A000040. So a(n+1)=1 if n=2^p-1 is a Mersenne prime A000668, as (n+1)!=(2!)^p*n! and n is prime. - Jonathan Sondow, Dec 15 2004
From Antti Karttunen, Dec 25 2018: (Start)
If n! = a! * x! * y! * ... * z!, with a > x >= y >= z, then A006530(n!) = A006530(a!) > A006530(x!). This follows because all rows in A115627 end with 1, that is, because all factorials >= 2 are in A102750.
If all the two-term solutions are of the form n! = a! * x! = b! * y! = ... = c! * z! (that is, all are products of two factorials larger than one), with a > x, b > y, ..., c > z, then a(n) = (a(x)+1 + a(y)+1 + ... + a(z)+1).
Values 0..5 occur for the first time at n = 1, 4, 10, 576, 13824, 69120.
In range 1..69120 differs from A322583 only at positions n = 1, 2, 9, 10 and 16.
(End)

			a(10) = 2 because 10! = 3! * 5! * 7! = 6! * 7! are the only two ways to write 10! as a product of smaller factorials > 1.
From _Antti Karttunen_, Dec 25 2018: (Start)
a(8) = 1 because 8! = 7! * (2!)^3.
a(9) = 1 because 9! = 7! * 3! * 3! * 2!.
a(16) = 2 because 16! = 15! * (2!)^4 = 14! * 5! * 2!.
a(144) = 2 because 144! = 143! * 4! * 3! = 143! * 3! * 3! * 2! * 2!.
a(576) = 3 because 576! = 575! * 4! * 4! = 575! * 4! * 3! * 2! * 2! = 575! * 3! * 3! * 2! * 2! * 2! * 2!.
a(720) = 2 because 720! = 719! * 6! = 719! * 5! * 3!.
a(3456) = 3 because 3456! = 3455! * 4! * 4! * 3! = 3455! * 4! * 3! * 3! * 2! * 2! = 3455! * 3! * 3! * 3! * 2! * 2! * 2! * 2!.
(End)

R. K. Guy, Unsolved Problems in Number Theory, B23.

Antti Karttunen, Table of n, a(n) for n = 1..69120
Eric Weisstein's World of Mathematics, Factorial Products
Index entries for sequences related to factorial numbers

Cf. A034878, A001013, A075082, A085604, A115627, A322583.

A034876aux(n, m, p) = if(1==n, 1, my(s=0); forstep(i=m, p, -1, my(f=i!); if(!(n%f), s += A034876aux(n/f, i, 2))); (s));
A034876(n) = if(1==n,0,A034876aux(n!, n-1, precprime(n))); \\ (Slow) - Antti Karttunen, Dec 24 2018

A322583aux(n, m) = if(1==n, 1, my(s=0); for(i=2, oo, my(f=i!); if(f>m, return(s)); if(!(n%f), s += A322583aux(n/f, f))));
memoA322583 = Map();
A322583(n) = { my(c); if(mapisdefined(memoA322583,n,&c), c, c = A322583aux(n,n); mapput(memoA322583,n,c); (c)); };
A034876aux(n, m, p) = if(1==n, 1, my(s=0); forstep(i=m, p, -1, my(f=i!); s += A322583(n/f)); (s));
A034876(n) = if(1==n, 0, A034876aux(n!, n-1, precprime(n))); \\ Antti Karttunen, Dec 25 2018

a(1) = 0; for n > 1, a(n) = Sum_{x=A007917(n)..n-1} A322583(n!/x!) when n is a composite, and a(n) = 0 when n is a prime. - Antti Karttunen, Dec 25 2018

Corrected by Jonathan Sondow, Dec 18 2004

A336496 Products of superfactorials (A000178).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 768, 1024, 1152, 1536, 1728, 2048, 2304, 3072, 3456, 4096, 4608, 6144, 6912, 8192, 9216, 12288, 13824, 16384, 18432, 20736, 24576, 27648, 32768, 34560, 36864, 41472, 49152, 55296

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

First differs from A317804 in having 34560, which is the first term with more than two distinct prime factors.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   48: {1,1,1,1,2}
   64: {1,1,1,1,1,1}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  512: {1,1,1,1,1,1,1,1,1}

A001013 is the version for factorials, with complement A093373.
A181818 is the version for superprimorials, with complement A336426.
A336497 is the complement.
A000178 lists superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A049711 is the minimum prime multiplicity in A000178.
A174605 is the maximum prime multiplicity in A000178.
A303279 counts prime factors of superfactorials.
A317829 counts factorizations of superprimorials.
A322583 counts factorizations into factorials.
A325509 counts factorizations of factorials into factorials.
Cf. A000142, A000720, A007489, A011371, A022559, A022915, A027423, A034878, A034876, A076954, A115627, A294068.

supfac[n_]:=Product[k!,{k,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[1000],facsusing[Rest[Array[supfac,30]],#]!={}&]

A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

Original entry on oeis.org

0, 3, 7, 13, 17, 18, 21, 31, 38, 43, 47, 57, 68, 73, 91, 99, 111, 117, 123, 132, 133, 157, 183, 211, 241, 242, 253, 255, 268, 273, 293, 302, 307, 313, 322, 327, 343, 381, 413, 421, 438, 443, 463, 487, 507, 515, 553, 557, 577, 593, 601, 651, 693, 697, 703, 707

Offset: 1

Pontus von Brömssen, Jun 19 2023

nonn

For the corresponding sequence for numbers of the form k^3+1 instead of k^2+1, the only terms known to me are 0 and 26, with 26^3+1 = (2^3+1)^2*(6^3+1).

			0 is a term because 0^2+1 = 1 equals the empty product.
3 is a term because 3^2+1 = 10 = 2*5 = (1^2+1)*(2^2+1).
38 is a term because 38^2+1 = 1445 = 5*17*17 = (2^2+1)*(4^2+1)^2. (This is the first term that requires more than two factors.)

David Trimas, Table of n, a(n) for n = 1..2260

Sequences that list those terms (or their indices or some other key) of a given sequence that are products of smaller terms of the same sequence (in other words, the nonprimitive terms of the multiplicative closure of the sequence):
  A018252 (A000027),
  A034878 (A000142),
  A068143 (A000217),
  A363492 (A000041),
  A363634 (A000959),
  A363635 (A003309),
  this sequence (A002522),
  A363637 (A005563),
  A363638 (A008864),
  A363750 (A006093),
  A364151 (A000292),
  A374372 (A000326),
  A374373 (A000384),
  A374374 (A002378),
  A374375 (A007531),
  A374500 (A000330),
  A374501 (A002411),
  A374502 (A002412).

g[lst_, p_] :=
  Module[{t, i, j},
   Union[Flatten[Table[t = lst[[i]]; t[[j]] = p*t[[j]];
      Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1],
    Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]];
multPartition[n_] :=
  Module[{i, j, p, e, lst = {{}}}, {p, e} =
    Transpose[FactorInteger[n]];
   Do[lst = g[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst];
output = Join[{0}, Flatten[Position[Table[
     test = Sqrt[multPartition[n^2 + 1][[2 ;; All]] - 1];
     Count[AllTrue[#, IntegerQ] & /@ test, True] > 0
     , {n, 707}], True]]]
(* David Trimas, Jul 23 2023 *)

cp's OEIS Frontend

A001013 Jordan-Polya numbers: products of factorial numbers A000142.

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A075082 Numbers n such that n! is a product of distinct factorials k!*l!*m!*... with k, l, m, etc. < n.

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A003135 n! is a nontrivial product of factorials. It is conjectured that the list is complete.

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A109095 Numbers N such that N! is the product of exactly two smaller factorials (larger than 1).

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A109096 Numbers n such that n! is the product of exactly three smaller factorials.

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A109097 Numbers n such that n! is the product of exactly four smaller factorials.

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A325509 Number of factorizations of n! into factorial numbers > 1.

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A034876 Number of ways to write n! as a product of smaller factorials each greater than 1.

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A075082 Numbers n such that n! is a product of distinct factorials k!l!m!*... with k, l, m, etc. < n.