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

%I A034878 #32 Feb 16 2025 08:32:37
%S A034878 1,4,6,8,9,10,12,16,24,32,36,48,64,72,96,120,128,144,192,216,240,256,
%T A034878 288,384,432,480,512,576,720,768,864,960,1024,1152,1296,1440,1536,
%U A034878 1728,1920,2048,2304,2592,2880,3072,3456,3840,4096,4320,4608,5040,5184
%N A034878 Numbers k such that k! can be written as the product of smaller factorials.
%C A034878 Except for the numbers 2, 9 and 10 this sequence is conjectured to be the same as A001013.
%C A034878 Every r! is a member for r>2, for (r!)! = (r!)*(r!-1)!. - _Amarnath Murthy_, Sep 11 2002
%C A034878 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
%C A034878 If there exists another term of this sequence not also in A001013, it must be >= 100000. - _Charlie Neder_, Oct 07 2018
%C A034878 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
%D A034878 R. K. Guy, Unsolved Problems in Number Theory, B23.
%H A034878 Charlie Neder, <a href="/A034878/b034878.txt">Table of n, a(n) for n = 1..222</a>
%H A034878 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FactorialProducts.html">Factorial Products</a>
%H A034878 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%e A034878 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.
%Y A034878 Cf. A000142, A075082, A001013.
%Y A034878 See also A359636, A359751.
%K A034878 easy,nonn,nice
%O A034878 1,2
%A A034878 _Erich Friedman_
%E A034878 More terms from _Jud McCranie_, Sep 13 2002
%E A034878 Edited by _Dean Hickerson_, Sep 17 2002