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

Author

Felix Fröhlich, Aug 25 2013

Keywords

• hard
• nonn
• more
• bref

Links

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

Crossrefs

The odd terms of A328497.

Cf. A099905, A136327.

Programs

Mathematica

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

PARI

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

PARI

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

Formula

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