A082180 Composite integers k such that binomial(2*k, k) == 2 (mod k).

4, 9, 25, 49, 121, 125, 169, 289, 343, 361, 418, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 17161, 18769, 19321

Offset: 1

Author

Benoit Cloitre, May 10 2003

Keywords

• nonn

Comments

Also, composite integers k such that A000108(k) == 2 (mod k).

It seems that the sequence contains all squares of primes and some cubes of odd primes. But it includes other terms as well, including 418 = 2*11*19 and 27173 = 29*937. [edited by Jon E. Schoenfield, Jul 31 2018]

By Wolstenholme's theorem, this sequence does contain all squares of primes and cubes of primes > 3^3, since for primes p > 3 we have binomial(2p^3, p^3) == binomial(2p^2, p^2) == binomial(2p, p) == binomial(2, 1) == 2 (mod p^3). See the link below. - Jianing Song, Aug 01 2018

Note that binomial(2*(n+1), n+1) = binomial(2*n, n) * (4 - 2/(n+1)), which could be used to find terms. - David A. Corneth, Aug 05 2018

Up to a(800) = 30946969, 2001341 = 787 * 2543 is the only further term which, like 418 and 27173, is neither a square nor a cube. - Giovanni Resta, Aug 08 2018

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..3301 (terms 1..193 from David A. Corneth, terms 194..1320 from Giovanni Resta)
• Wikipedia, Wolstenholme's theorem

Crossrefs

Cf. A000108, A246131, A328497.

Programs

GAP

Filtered([1..1000],n->not IsPrime(n) and Binomial(2*n,n) mod n =2); # Muniru A Asiru, Aug 01 2018

Maple

select(n-> not isprime(n) and modp(binomial(2*n,n),n)=2,[$1..10000]); # Muniru A Asiru, Aug 01 2018

Mathematica

nn=20000;With[{comps=Complement[Range[nn],Prime[Range[PrimePi[nn]]]]}, Select[ comps,Mod[Binomial[2#,#],#]==2&]] (* Harvey P. Dale, May 24 2012 *)
Select[Range@ 20000, CompositeQ@# && Mod[Binomial[2 #, #], #] == 2 &] (* Robert G. Wilson v, Aug 01 2018 *)

PARI

forcomposite(c=1, 2e4, if(Mod(binomial(2*c, c), c)==2, print1(c, ", "))) \\ Felix Fröhlich, Jul 30 2018

PARI

upto(n) = {my(binomp = 2, res = List()); for(t = 2, n, binomp *= (4 - 2/t);
if(!isprime(t) && binomp % t == 2, listput(res, t))); res} \\ David A. Corneth, Aug 05 2018

Extensions

More terms from John W. Layman, Jun 09 2004