A205947 - OEIS

%I #35 Jul 20 2015 16:43:24

%S 561,2465,62745,162401,656601,1909001,5444489,11921001,19384289,

%T 26719701,45318561,84350561,151530401,174352641,221884001,230996949,

%U 275283401,434932961,662086041,684106401,689880801,710382401

%N Carmichael numbers not congruent to 1 modulo 6.

%C These numbers are very sparse; most Carmichael numbers are 1 mod 6. - _Charles R Greathouse IV_, May 02 2012

%C Not known to be infinite, see Matomäki. - _Charles R Greathouse IV_, Jun 13 2012

%C From _Robert Israel_, Jul 20 2015: (Start)

%C Now known to be infinite, see Wright.

%C 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 Charles R Greathouse IV, <a href="/A205947/b205947.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 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 korselt:= proc(n) uses numtheory; local p;

%p   if isprime(n) or not issqrfree(n) then return false fi;

%p   for p in factorset(n) do

%p      if n-1 mod (p-1) <> 0 then return false fi

%p   od;

%p   true

%p end proc:

%p select(korselt, [seq(seq(6*i+j,j=[3,5]),i=1..10^5)]); # _Robert Israel_, Jul 20 2015

%t Select[Range[100000], !PrimeQ[#] && IntegerQ[(#-1)/CarmichaelLambda[#]] && !Mod[#,6]==1&]

%o (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 list(lim)={

%o   my(v=List(),p=2);

%o   forstep(n=561,lim,[12,6],

%o     if(Korselt(n),listput(v,n))

%o   );

%o   forprime(q=3,lim,

%o     forstep(n=p+if(p%6<5,4,6),q-2,6,

%o       if(Korselt(n),listput(v,n))

%o     );

%o     p=q

%o   );

%o   vecsort(Vec(v))

%o }; \\ _Charles R Greathouse IV_, Apr 25 2012

%Y Cf. A002997, A045410, A258801.

%K nonn

%O 1,1

%A _José María Grau Ribas_, Feb 02 2012