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

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

Author

Amarnath Murthy, Sep 11 2002

Keywords

• nonn

Comments

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

Examples

			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.
		

References

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

Links

• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A034878, A001013, A058295.

Programs

Mathematica

(* 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 *)

Extensions

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