

A075082


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


16



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
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OFFSET

1,2


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 (JordanPolya 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, n1.  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


REFERENCES

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


LINKS

Table of n, a(n) for n=1..38.
Index entries for sequences related to factorial numbers.


EXAMPLE

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.


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 *)


CROSSREFS

Cf. A034878, A001013, A058295.
Sequence in context: A315123 A315124 A063214 * A101503 A102148 A107407
Adjacent sequences: A075079 A075080 A075081 * A075083 A075084 A075085


KEYWORD

nonn


AUTHOR

Amarnath Murthy, Sep 11 2002


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


STATUS

approved



