A246133 a(n) = (binomial(2n, n) - 2) mod n^3.

0, 4, 18, 4, 0, 58, 0, 68, 504, 754, 0, 1562, 0, 2062, 2518, 580, 0, 922, 0, 818, 6535, 7990, 0, 12058, 250, 4398, 2691, 10358, 0, 12422, 0, 16964, 10666, 29482, 3680, 42818, 0, 41158, 19791, 13618, 0, 54430, 0, 71942, 40993, 73006, 0, 12058, 3430, 122254, 98278, 127494, 0

Offset: 1

Author

Stanislav Sykora, Aug 16 2014

Keywords

• nonn

Comments

When e=3, the numbers binomial(2n, n) - 2 mod n^e are 0 whenever n is a prime greater than 3 (Wolstenholme's theorem; see A246130 for introductory comments). No composite number n for which a(n)=0 was found up to n=431500 (conjecture: there are none, and a(n)=0 for n>3 is a deterministic primality test).

Examples

			a(7)= (binomial(14,7)-2) mod 7^3 = (3432-2) mod 343 = 10*343 mod 343 = 0.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..10000
• Wikipedia, Wolstenholme's theorem

Crossrefs

Cf. A000984, A246130 (e=1), A246132 (e=2), A246134 (e=4).

Programs

Maple

seq(binomial(2*n,n)-2 mod n^3, n=1..100); # Robert Israel, Aug 17 2014

Mathematica

Table[Mod[Binomial[2 n, n] - 2, n^3], {n, 60}] (* Wesley Ivan Hurt, May 25 2024 *)

PARI

a(n) = (binomial(2*n,n)-2)%n^3

Formula

For any prime p>3, a(p)=0.