A364152 Least n-simplex number (i.e., number of the form C(m,n) = binomial(m,n), m >= n), that can be written as a product of two or more smaller n-simplex numbers, or 0 if no such number exists.

4, 36, 560, 20475, 126

Offset: 1

Author

Pontus von Brömssen, Jul 15 2023

Keywords

• nonn
• more

Comments

When n = k^2+3*k+1 is in A028387, C(n+k+3,n) = C(n+1,n) * C(n+k+1,n), so 0 != a(n) <= C(n+k+3,n). It appears that equality holds (verified for 0 <= k <= 100). In particular, a(11) = C(16,11) = 4368, a(19) = C(25,19) = 177100, a(29) = C(36,29) = 8347680, a(41) = C(49,41) = 450978066, ... .

a(34) = 4923689695575 = C(50,34) = C(35,34)*C(47,34).

a(6) > 10^29 (unless a(6) = 0). - Pontus von Brömssen, Jul 14 2024

Examples

			a(1) =     4 = C( 4, 1) = C(2,1) * C(2,1).
a(2) =    36 = C( 9, 2) = C(4,2)^2.
a(3) =   560 = C(16, 3) = C(5,3) * C(8,3). (Also, C(16,3) = C(4,3)^2 * C(7,3)).
a(4) = 20475 = C(28, 4) = C(6,4) * C(15,4).
a(5) =   126 = C( 9, 5) = C(6,5) * C(7,5).
		

Crossrefs

a(1)-a(3) are the first terms greater than 1 in A018252, A068143, and A364151, respectively.

Cf. A028387.