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

Author

Erich Friedman

Keywords

• easy
• nonn
• nice

Comments

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

Examples

			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.
		

References

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

Links

• 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

Crossrefs

Cf. A000142, A075082, A001013.

See also A359636, A359751.

Extensions

More terms from Jud McCranie, Sep 13 2002

Edited by Dean Hickerson, Sep 17 2002