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

1,1

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

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

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&]

Prog

(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

Crossrefs

Cf. A002997, A045410, A258801.

Sequence in context: A217465 A097130 A110889 * A290692 A293622 A322130

Adjacent sequences: A205944 A205945 A205946 * A205948 A205949 A205950

Keyword

nonn

Author

José María Grau Ribas, Feb 02 2012

Status

approved