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%I #15 May 25 2024 23:02:54
%S 0,4,18,4,0,58,0,68,504,754,0,1562,0,2062,2518,580,0,922,0,818,6535,
%T 7990,0,12058,250,4398,2691,10358,0,12422,0,16964,10666,29482,3680,
%U 42818,0,41158,19791,13618,0,54430,0,71942,40993,73006,0,12058,3430,122254,98278,127494,0
%N a(n) = (binomial(2n, n) - 2) mod n^3.
%C 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).
%H Stanislav Sykora, <a href="/A246133/b246133.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Wolstenholme%27s_theorem">Wolstenholme's theorem</a>
%F For any prime p>3, a(p)=0.
%e a(7)= (binomial(14,7)-2) mod 7^3 = (3432-2) mod 343 = 10*343 mod 343 = 0.
%p seq(binomial(2*n,n)-2 mod n^3, n=1..100); # _Robert Israel_, Aug 17 2014
%t Table[Mod[Binomial[2 n, n] - 2, n^3], {n, 60}] (* _Wesley Ivan Hurt_, May 25 2024 *)
%o (PARI) a(n) = (binomial(2*n,n)-2)%n^3
%Y Cf. A000984, A246130 (e=1), A246132 (e=2), A246134 (e=4).
%K nonn
%O 1,2
%A _Stanislav Sykora_, Aug 16 2014
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