|
%I #64 Jun 04 2024 15:36:26
%S 6,36,240,2520,30240,443520,6652800,133056000,2075673600,58118860800,
%T 1270312243200,29640619008000,844757641728000,25342729251840000,
%U 810967336058880000,27978373094031360000,1077167364120207360000,43086694564808294400000,1499416970855328645120000
%N a(n) is the least integer for which there exist two disjoint sets of n positive integers each, all distinct, for which the product of the integers in either set is a(n).
%C This is also the least integer that can be represented as the product of the integers > 1 in two disjoint sets, one having n terms and the other having n-1 terms.
%C From _Jon E. Schoenfield_, May 12 2024: (Start)
%C For n >= 2, let b(n) be the square root of the smallest square that can be expressed as the product of 2*n distinct positive integers; then a(n) >= b(n).
%C Conjecture: for every n >= 2, a(n) = b(n). (End)
%H Zhao Hui Du, <a href="/A354457/b354457.txt">Table of n, a(n) for n = 2..28</a>
%H Shouwen Wang, <a href="https://bbs.emath.ac.cn/forum.php?mod=viewthread&tid=19390&page=4#pid100355">Discussion on Chinese BBS on A354457</a>
%e From _Jinyuan Wang_, May 31 2022: (Start)
%e For n=2, 6 = 1*6 = 2 * 3.
%e For n=3, 36 = 1*4*9 = 2 * 3 * 6.
%e For n=4, 240 = 1*3*8*10 = 2 * 4 * 5 * 6.
%e For n=5, 2520 = 1*2*9*10*14 = 3 * 4 * 5 * 6 * 7.
%e For n=6, 30240 = 1*2*6*10*14*18 = 3 * 4 * 5 * 7 * 8 * 9.
%e For n=7, 443520 = 1*2*5*9*14*16*22 = 3 * 4 * 6 * 7 * 8 *10 *11.
%e For n=8, 6652800 = 1*2*3*12*14*15*20*22 = 4 * 5 * 6 * 7 * 8 * 9 *10 *11. (End)
%e From _Zhao Hui Du_, May 11 2024: (Start)
%e For n=9, 133056000 = 1*2*3*9*14*16*20*22*25 = 4*5*6*7*8*10*11*12*15.
%e For n=10, 2075673600 = 1*2*3*7*15*16*18*20*22*26 = 4*5*6*8*9*10*11*12*13*14. (End)
%Y Cf. A001055, A025487, A354697.
%K nonn
%O 2,1
%A _Andy Niedermaier_, May 30 2022
%E a(7)-a(8) from _Jinyuan Wang_, May 31 2022
%E a(9)-a(10) from _Zhao Hui Du_, May 11 2024
%E a(11)-a(20) from _Jon E. Schoenfield_, May 11 2024
| 