A328497 -id:A328497 - cp's OEIS frontend

A228562 Composite numbers k that are not prime powers such that binomial(2k-1, k-1) is congruent to 1 (mod k).

27173, 2001341, 16024189487

Offset: 1

Felix Fröhlich, Aug 25 2013

Richard J. McIntosh, On the converse of Wolstenholme's Theorem, Acta Arithmetica, 71 (1995), 381-389.

The odd terms of A328497.

Select[Range[30000], PrimeNu[#] > 1 && Mod[Binomial[2# - 1, # - 1], #] == 1 &] (* Alonso del Arte, May 11 2014 *)

N=10^9; for(n=2, N, if(Mod(binomial(2*n-1, n-1), n)==1 && !ispower(n) && !isprime(n), print1(n, ", "))); \\ Felix Fröhlich, May 11 2014

vp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=my(f=factor(n)[,1],G); if(#f==1, return(0)); for(i=1,#f, if(vp(2*n-1,f[i]) > vp(n,f[i])+vp(n-1,f[i]), return(0))); G=prod(i=1,#f,f[i]^(log(n)\log(f[i]))); prod(i=n+1,2*n-1, i/gcd(i,G), Mod(1,n))/prod(i=2,n-1, i/gcd(i,G), Mod(1,n))==1
forcomposite(n=4,1e9, if(is(n), print1(n", "))) \\ Charles R Greathouse IV, May 12 2014

A099905(a(n)) = 1. - Jonathan Sondow, Jan 24 2016

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

Original entry on oeis.org

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

Benoit Cloitre, May 10 2003

nonn

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

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

Cf. A000108, A246131, A328497.

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

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

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 *)

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

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

More terms from John W. Layman, Jun 09 2004

A084699 Composite integers j such that binomial(2*j,j) == 2^j (mod j).

Original entry on oeis.org

12, 30, 56, 424, 992, 16256, 58288, 119984, 356992, 1194649, 9973504, 12327121, 13141696, 22891184, 67100672, 233850649

Offset: 1

Benoit Cloitre, Oct 15 2003

If p is prime, binomial(2*p,p) == 2^p (mod p).
a(17) > 10^9.
From Gabriel Guedes and Ricardo Machado, Nov 16 2023: (Start)
Theorem. Let j = (2^k)*p, where p is an odd prime and k is in N; then binomial(2*j,j) == 2^j (mod j) if and only if p satisfies the following conditions:
a) p divides binomial(2^(k+1),2^k) - 2^(2^k);
b) p has at least k 1's in its binary expansion.
Theorem. If m is an even perfect number then j = 2m satisfies the congruence binomial(2*j,j) == 2^j (mod j). See A000396.
Theorem. Let j = p^2 with p a prime number. Then p is a Wieferich prime if and only if binomial(2*j,j) == 2^j (mod j). See A001220. (End)
Contains 17179738112 and 274877382656 (from Guedes-Machado paper). - Michael De Vlieger, Nov 22 2023
Contains 3386741824, 750984028672, 33029195197184, 1145067923695616, 422612863956511744. - Ricardo Machado, Nov 23 2023
Contains 84385517065596416, 62648180117928433664, 273984397779878971648, 36506097537257040703232. - Max Alekseyev, Dec 07 2023

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems, sect. III: Binomial coefficients modulo integers, binomod.gp (V. 1.4, 11/2015).
Gabriel Guedes and Ricardo Machado, Perfect numbers, Wieferich primes and the solutions of binomial(2n, n) congruent to 2^n mod n, Notes Num. Theor. Disc. Math. (NNTDM 2023) Vol. 29, No. 4, 705-712.

Contains A139256 as a subsequence.

Cf. A015910, A059288, A080469, A082180, A109760, A109769, A228562, A328497.

lista(nn) = {forcomposite(n=1, nn, if (binomod(2*n, n, n) == Mod(2, n)^n, print1(n, ", ")));} \\ Michel Marcus, Dec 06 2013 and Dec 03 2023

More terms from David Wasserman, Jan 03 2005

a(11)-a(16) from Max Alekseyev, Aug 05 2011

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A228562 Composite numbers k that are not prime powers such that binomial(2k-1, k-1) is congruent to 1 (mod k).

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

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A084699 Composite integers j such that binomial(2*j,j) == 2^j (mod j).

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