A354457 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).

6, 36, 240, 2520, 30240, 443520, 6652800, 133056000, 2075673600, 58118860800, 1270312243200, 29640619008000, 844757641728000, 25342729251840000, 810967336058880000, 27978373094031360000, 1077167364120207360000, 43086694564808294400000, 1499416970855328645120000

Offset: 2

Author

Andy Niedermaier, May 30 2022

Keywords

• nonn

Comments

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.

From Jon E. Schoenfield, May 12 2024: (Start)

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).

Conjecture: for every n >= 2, a(n) = b(n). (End)

Examples

			From _Jinyuan Wang_, May 31 2022: (Start)
For n=2,       6 = 1*6                  = 2 * 3.
For n=3,      36 = 1*4*9                = 2 * 3 * 6.
For n=4,     240 = 1*3*8*10             = 2 * 4 * 5 * 6.
For n=5,    2520 = 1*2*9*10*14          = 3 * 4 * 5 * 6 * 7.
For n=6,   30240 = 1*2*6*10*14*18       = 3 * 4 * 5 * 7 * 8 * 9.
For n=7,  443520 = 1*2*5*9*14*16*22     = 3 * 4 * 6 * 7 * 8 *10 *11.
For n=8, 6652800 = 1*2*3*12*14*15*20*22 = 4 * 5 * 6 * 7 * 8 * 9 *10 *11. (End)
From _Zhao Hui Du_, May 11 2024: (Start)
For n=9, 133056000 = 1*2*3*9*14*16*20*22*25 = 4*5*6*7*8*10*11*12*15.
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)
		

Links

• Zhao Hui Du, Table of n, a(n) for n = 2..28
• Shouwen Wang, Discussion on Chinese BBS on A354457

Crossrefs

Cf. A001055, A025487, A354697.

Extensions

a(7)-a(8) from Jinyuan Wang, May 31 2022

a(9)-a(10) from Zhao Hui Du, May 11 2024

a(11)-a(20) from Jon E. Schoenfield, May 11 2024