A205947 Carmichael numbers not congruent to 1 modulo 6.

561, 2465, 62745, 162401, 656601, 1909001, 5444489, 11921001, 19384289, 26719701, 45318561, 84350561, 151530401, 174352641, 221884001, 230996949, 275283401, 434932961, 662086041, 684106401, 689880801, 710382401

Offset: 1

Author

José María Grau Ribas, Feb 02 2012

Keywords

• nonn

Comments

These numbers are very sparse; most Carmichael numbers are 1 mod 6. - Charles R Greathouse IV, May 02 2012

Not known to be infinite, see Matomäki. - Charles R Greathouse IV, Jun 13 2012

From Robert Israel, Jul 20 2015: (Start)

Now known to be infinite, see Wright.

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)

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Kaisa Matomäki, Carmichael numbers in arithmetic progressions, Journal of the Australian Mathematical Society 94:2 (2013), pp. 268-275.
• T. Wright, Infinitely many Carmichael numbers in arithmetic progressions, Bull. London Math. Soc. (2013) 45 (5): 943-952. arXiv:1212.5850

Crossrefs

Cf. A002997, A045410, A258801.

Programs

Maple

korselt:= proc(n) uses numtheory; local p;
  if isprime(n) or not issqrfree(n) then return false fi;
  for p in factorset(n) do
     if n-1 mod (p-1) <> 0 then return false fi
  od;
  true
end proc:
select(korselt, [seq(seq(6*i+j,j=[3,5]),i=1..10^5)]); # Robert Israel, Jul 20 2015

Mathematica

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

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
list(lim)={
  my(v=List(),p=2);
  forstep(n=561,lim,[12,6],
    if(Korselt(n),listput(v,n))
  );
  forprime(q=3,lim,
    forstep(n=p+if(p%6<5,4,6),q-2,6,
      if(Korselt(n),listput(v,n))
    );
    p=q
  );
  vecsort(Vec(v))
}; \\ Charles R Greathouse IV, Apr 25 2012

Formula

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