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A205947 Carmichael numbers not congruent to 1 modulo 6.

%I A205947 #35 Jul 20 2015 16:43:24
%S A205947 561,2465,62745,162401,656601,1909001,5444489,11921001,19384289,
%T A205947 26719701,45318561,84350561,151530401,174352641,221884001,230996949,
%U A205947 275283401,434932961,662086041,684106401,689880801,710382401
%N A205947 Carmichael numbers not congruent to 1 modulo 6.
%C A205947 These numbers are very sparse; most Carmichael numbers are 1 mod 6. - _Charles R Greathouse IV_, May 02 2012
%C A205947 Not known to be infinite, see Matomäki. - _Charles R Greathouse IV_, Jun 13 2012
%C A205947 From _Robert Israel_, Jul 20 2015: (Start)
%C A205947 Now known to be infinite, see Wright.
%C A205947 No member of this sequence is divisible by any prime of the form 6k+1, hence all prime factors for this sequence are members of A045410. (End)
%H A205947 Charles R Greathouse IV, <a href="/A205947/b205947.txt">Table of n, a(n) for n = 1..10000</a>
%H A205947 Kaisa Matomäki, <a href="http://users.utu.fi/ksmato/papers/CarmichaelAPs.pdf">Carmichael numbers in arithmetic progressions</a>, Journal of the Australian Mathematical Society 94:2 (2013), pp. 268-275.
%H A205947 T. Wright, <a href="http://dx.doi.org/10.1112/blms/bdt013">Infinitely many Carmichael numbers in arithmetic progressions</a>, Bull. London Math. Soc. (2013) 45 (5): 943-952. <a href="http://arxiv.org/abs/1212.5850">arXiv:1212.5850</a>
%F A205947 Wright shows that there are at least x^(K/(log log log x)^2) terms up to x, for an explicitly computable (though not computed) constant K. - _Charles R Greathouse IV_, Jul 20 2015
%p A205947 korselt:= proc(n) uses numtheory; local p;
%p A205947   if isprime(n) or not issqrfree(n) then return false fi;
%p A205947   for p in factorset(n) do
%p A205947      if n-1 mod (p-1) <> 0 then return false fi
%p A205947   od;
%p A205947   true
%p A205947 end proc:
%p A205947 select(korselt, [seq(seq(6*i+j,j=[3,5]),i=1..10^5)]); # _Robert Israel_, Jul 20 2015
%t A205947 Select[Range[100000], !PrimeQ[#] && IntegerQ[(#-1)/CarmichaelLambda[#]] && !Mod[#,6]==1&]
%o A205947 (PARI) Korselt(n,f=factor(n))=for(i=1,#f[,1],if(f[i,2]>1||(n-1)%(f[i,1]-1),return(0)));1
%o A205947 list(lim)={
%o A205947   my(v=List(),p=2);
%o A205947   forstep(n=561,lim,[12,6],
%o A205947     if(Korselt(n),listput(v,n))
%o A205947   );
%o A205947   forprime(q=3,lim,
%o A205947     forstep(n=p+if(p%6<5,4,6),q-2,6,
%o A205947       if(Korselt(n),listput(v,n))
%o A205947     );
%o A205947     p=q
%o A205947   );
%o A205947   vecsort(Vec(v))
%o A205947 }; \\ _Charles R Greathouse IV_, Apr 25 2012
%Y A205947 Cf. A002997, A045410, A258801.
%K A205947 nonn
%O A205947 1,1
%A A205947 _José María Grau Ribas_, Feb 02 2012