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
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Feb 02 2012
STATUS
approved