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%I #16 Oct 02 2012 02:36:54
%S 1729,2821,41041,63973,101101,126217,172081,188461,294409,399001,
%T 488881,512461,670033,748657,838201,852841,997633,1033669,1050985,
%U 1082809,1461241,2100901,2113921,2628073,4463641,4909177,7995169,8341201,8719309,9890881
%N Carmichael numbers of the form C = p*(2p-1)*(n*(2p-2)+p), where p and 2p-1 are prime numbers.
%C Conjecture: Any Carmichael number C divisible by p and 2p-1 (where p and 2p-1 are prime numbers) can be written as C = p*(2p-1)*(n*(2p-2)+p).
%C Checked for the first 30 Carmichael numbers divisible by p and 2p-1.
%C Note: we can see how easy is to obtain Carmichael numbers with this formula:
%C for n = 1 we get p*(2p-1)*(3p-2) and Carmichael numbers 1729, 172081, 294409 etc.
%C for n = 2 we get p*(2p-1)*(5p-4) and Carmichael numbers 2821, 63973, 488881 etc.
%C for n = 3 we get p*(2p-1)*(7p-6) and Carmichael numbers 399001, 53711113 etc.
%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>
%o (PARI) Korselt(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,2]>1||(n-1)%(f[i,1]-1),return(0)));1
%o list(lim)=my(v=List(),q,t);forprime(p=3,round(solve(x=1,lim,6*x^3-7*x^2+2*x-lim)),for(n=1,(lim\(2*p^2-p)-p)\(2*p-2),if(isprime(q=2*p-1)&&Korselt(t=p*q*(n*q-n+p)),listput(v,t))));vecsort(Vec(v),,8)
%o \\ _Charles R Greathouse IV_, Oct 02 2012
%K nonn
%O 1,1
%A _Marius Coman_, Apr 18 2012
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