A099905 - OEIS

%I #20 Sep 08 2022 08:45:15

%S 0,1,1,3,1,0,1,3,1,8,1,2,1,10,0,3,1,12,1,10,3,14,1,6,1,16,10,0,1,2,1,

%T 3,21,20,21,26,1,22,10,10,1,0,1,24,0,26,1,30,1,28,27,48,1,30,16,44,48,

%U 32,1,48,1,34,6,35,35,0,1,18,33,20,1,18,1,40,60,16,0,72,1,10,10,44,1,56,75

%N a(n) = binomial(2n-1, n-1) mod n.

%C For p prime, a(p)=1. For n in A058008, a(n)=0.

%C For n the square of a prime p>=3 or the cube of a prime p>=5, a(n)=1. - _Franz Vrabec_, Mar 26 2008

%C For n in A228562, a(n)=1. - _Felix Fröhlich_, Oct 17 2015

%H Felix Fröhlich, <a href="/A099905/b099905.txt">Table of n, a(n) for n = 1..10000</a>

%H R. J. McIntosh, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7144.pdf">On the converse of Wolstenholme's theorem</a>, Acta Arithmetica 71 (4): 381-389, (1995).

%e a(11) = 352716 mod 11 = 1.

%p A099905:=n->binomial(2*n-1,n-1) mod n: seq(A099905(n), n=1..100); # _Wesley Ivan Hurt_, Oct 17 2015

%t Table[Mod[Binomial[2n-1,n-1],n],{n,90}] (* _Harvey P. Dale_, Dec 12 2011 *)

%o (PARI) a(n) = lift(Mod(binomial(2*n-1, n-1), n)) \\ _Felix Fröhlich_, Oct 17 2015

%o (Magma) [Binomial(2*n-1,n-1) mod n : n in [1..100]]; // _Wesley Ivan Hurt_, Oct 17 2015

%Y Cf. A058008, A088218, A099906, A099907, A099908, A228562.

%K nonn,easy

%O 1,4

%A _Henry Bottomley_, Oct 29 2004