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A082180 Composite integers k such that binomial(2*k, k) == 2 (mod k). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Giovanni Resta, Table of n, a(n) for n = 1..1320 (terms < 10^8, first 193 terms from David A. Corneth)

Wikipedia, Wolstenholme's theorem

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 *)

PROG

(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

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

CROSSREFS

Cf. A000108, A246131, A328497.

Sequence in context: A158144 A247136 A158145 * A246131 A068999 A179707

Adjacent sequences:  A082177 A082178 A082179 * A082181 A082182 A082183

KEYWORD

nonn

AUTHOR

Benoit Cloitre, May 10 2003

EXTENSIONS

More terms from John W. Layman, Jun 09 2004

STATUS

approved

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Last modified August 8 23:02 EDT 2020. Contains 336300 sequences. (Running on oeis4.)