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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.

%I A075082 #21 Mar 17 2015 00:53:34
%S A075082 1,6,10,12,16,24,48,120,144,240,288,720,1440,2880,4320,5040,5760,8640,
%T A075082 10080,17280,30240,34560,40320,60480,80640,86400,103680,120960,172800,
%U A075082 207360,241920,362880,483840,518400,604800,725760,967680,1036800
%N A075082 Numbers n such that n! is a product of distinct factorials k!*l!*m!*... with k, l, m, etc. < n.
%C A075082 r! is a member for r>2, since (r!)! = (r!)*(r!-1)!.
%C A075082 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
%C A075082 May be the same as A058295 except for 2, 10 and 16. - _Jud McCranie_, Jun 13 2005
%C A075082 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
%C A075082 Except for 1, 10 & 16, all the members are of the form immediately above. - _Robert G. Wilson v_, Jan 27 2006
%C A075082 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
%C A075082 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
%D A075082 R. K. Guy, Unsolved Problems in Number Theory, B23.
%H A075082 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers.</a>
%e A075082 1! = 0!, 6! = 5!*3!, 10! = 7!*6!, 12! = 11!*3!*2!, 16! = 14!*5!*2!,
%e A075082 24! = 23!*4!, 48! = 47!*4!*2!, 120! = 119!*5!, 144! = 143! *4!*3!,
%e A075082 240! = 239!*5!*2!, 288! = 287!*4!*3!*2!, 720! = 719!*6!,
%e A075082 1440! = 1439!6!*2!, etc.
%t A075082 (* 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 *)
%Y A075082 Cf. A034878, A001013, A058295.
%K A075082 nonn
%O A075082 1,2
%A A075082 _Amarnath Murthy_, Sep 11 2002
%E A075082 Corrected and extended by _Jud McCranie_, Sep 13 2002
%E A075082 More terms from _Jud McCranie_, Jun 13 2005
%E A075082 a(25)-a(39) proposed by _Robert G. Wilson v_, Jun 18 2005, confirmed by _David Wasserman_, Dec 30 2005