A075082 -id:A075082 - cp's OEIS frontend

A034878 Numbers k such that k! can be written as the product of smaller factorials.

1, 4, 6, 8, 9, 10, 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

Offset: 1

Erich Friedman

Except for the numbers 2, 9 and 10 this sequence is conjectured to be the same as A001013.
Every r! is a member for r>2, for (r!)! = (r!)*(r!-1)!. - Amarnath Murthy, Sep 11 2002
By Murthy's trick, if k>2 is a product of factorials then k is a term. So half of the above conjecture is true: A001013 is a subsequence except for the number 2. - Jonathan Sondow, Nov 08 2004
If there exists another term of this sequence not also in A001013, it must be >= 100000. - Charlie Neder, Oct 07 2018
An additional term of this sequence not in A001013 must be > 5000000. Can it be shown that no such terms exist using results on consecutive smooth numbers? - Charlie Neder, Jan 14 2019

			1! = 0! (or, 1! is the empty product), 4! = 2!*2!*3!, 6! = 3!*5!, 8! = (2!)^3*7!, 9! = 2!*3!*3!*7!, 10! = 6!*7!, etc.

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

Charlie Neder, Table of n, a(n) for n = 1..222
Eric Weisstein's World of Mathematics, Factorial Products
Index entries for sequences related to factorial numbers

Cf. A000142, A075082, A001013.

A058295 Products of distinct factorials.

Original entry on oeis.org

1, 2, 6, 12, 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, 1209600

Offset: 1

Leroy Quet, Dec 07 2000

nonn

(A075082(n)!)^2 is a member for n>0, for example, (6!)^2=6!*5!*3!. Factorials A000142 and superfactorials A000178 (without their first terms), double-superfactorials A098694 and product-of-next-n-factorials A074319 are all subsequences. Products-of-factorials A001013 is a supersequence. - Jonathan Sondow, Dec 18 2004
A000197(n)^2 is a member for n > 2, as ((n!)!)^2 = (n!)!*n!*(n!-1)!. - Jonathan Sondow, Dec 21 2004
Erdős & Graham show that there are exp((1+o(1))n log log n / log n) members of this sequence using no factorials above n.

			288 is included because 288 = 2! * 3! * 4!.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355.
Index entries for sequences related to factorial numbers

Cf. A075082, A000142, A000178, A098694, A074319, A001013, A000197.

k=10; m=1; With[{p=With[{s=Subsets[Table[n!, {n, 2, k}]]}, Sort[Table[Apply[Times, s[[n]]], {n, Length[s]}]]]}, While[p[[m]]<(k+1)!, m++ ]; Union[Take[p, m-1]]] (* Jonathan Sondow *)

list(lim)=my(v=List([1]),n=1,t=1);while((t=n++!)<=lim,for(i=1,#v,if(v[i]*t<=lim,listput(v,v[i]*t))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Mar 26 2012

Corrected by Jonathan Sondow, Dec 18 2004

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

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

A109098 Numbers n such that n! is the product of exactly 5 smaller factorials (greater than 1).

Original entry on oeis.org

16, 48, 144, 192, 432, 576, 960, 1296, 1728, 2304, 2880, 5184, 5760, 6912, 8640, 11520, 17280, 20736, 25920, 27648, 34560, 40320, 51840, 57600, 69120, 82944

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			144! = 2! * 2! * 3! * 3! * 143!, so 144 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097.

A109100 Numbers n such that n! is the product of exactly 2 smaller factorials (greater than 1).

Original entry on oeis.org

10, 16, 24, 48, 96, 144, 192, 288, 384, 720, 768, 864, 1440, 1536, 1728, 3072, 4320, 5184, 6144, 8640, 10368, 12288, 24576, 25920, 31104, 40320, 49152, 51840, 62208, 80640, 86400, 98304

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			10! = 3! * 5! * 7! = 6! * 7!, so 10 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109100, A109101, A109102, A109103.

A109101 Numbers n such that n! is the product of exactly 3 smaller factorials (greater than 1).

Original entry on oeis.org

576, 1152, 2304, 2880, 3456, 4608, 5760, 6912, 9216, 11520, 18432, 20736, 23040, 36864, 41472, 46080, 73728, 92160

Offset: 1

Jud McCranie, Jun 19 2005

nonn

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109102, A109103.

cp's OEIS Frontend

A034878 Numbers k such that k! can be written as the product of smaller factorials.

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A058295 Products of distinct factorials.

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

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A109098 Numbers n such that n! is the product of exactly 5 smaller factorials (greater than 1).

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A109100 Numbers n such that n! is the product of exactly 2 smaller factorials (greater than 1).

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