%I #36 Dec 15 2023 18:50:48
%S 27173,2001341,16024189487
%N Composite numbers k that are not prime powers such that binomial(2k-1, k-1) is congruent to 1 (mod k).
%H Richard 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 (1995), 381-389.
%F A099905(a(n)) = 1. - _Jonathan Sondow_, Jan 24 2016
%t Select[Range[30000], PrimeNu[#] > 1 && Mod[Binomial[2# - 1, # - 1], #] == 1 &] (* _Alonso del Arte_, May 11 2014 *)
%o (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
%o (PARI) vp(n,p)=my(s); while(n\=p, s+=n); s
%o 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
%o forcomposite(n=4,1e9, if(is(n), print1(n", "))) \\ _Charles R Greathouse IV_, May 12 2014
%Y The odd terms of A328497.
%Y Cf. A099905, A136327.
%K hard,nonn,more,bref
%O 1,1
%A _Felix Fröhlich_, Aug 25 2013