%I #33 Dec 07 2014 16:28:37

%S 561,41041,75361,101101,656601,658801,852841,2100901,2704801,4903921,

%T 6049681,6313681,8341201,9890881,10837321,10877581,11205601,18162001,

%U 26932081,27062101,29020321,33302401,37167361,37280881,40430401,43286881,67994641,72108421,76595761,77826001,88689601,93614521,100427041,104404861,105869401

%N Carmichael numbers divisible by 11.

%C Conjecture: Any Carmichael number C divisible by 11 and not divisible by 3 can be written in one of the following four forms: C = 990k+11; C = 990k+11^2; C = 990k+11*41 or C = 990k+11*71, where k is a natural number.

%C In other words, C mod 990 can be, for C divisible by 11 and not divisible by 3, just 11; 11^2; 11*41 or 11*71.

%C For comparison, we issue the following related conjectures:

%C (1) Any Carmichael number C of the form 10n+1 divisible by 7 and not divisible by 3 and 5 can be written in one of the following three forms: C = 630k+7*13; C = 630k+7*43; C = 7*73.

%C (2) Any Carmichael number C of the form 10n+3, 10n+7 or 10n+9 divisible by 7 and not divisible by 3 and 5 can be written as C = 630k+7^3, C = 630k+7*31 or C = 630k+7*67.

%C (3) Any Carmichael number C divisible same time with 7 and 11 can be written as C = 7*11*90*k + 7*11*n, where n can be 23, 53 or 83.

%C Note: there exist Carmichael numbers that can be written as C = p*(90k+1) for every p prime divisor of C.

%C Example:

%C C = 197531244744661; C mod 90*1531 = 1531; C mod 90*3061 = 3061; C mod 90*4591 = 4591; C mod 90*9181 = 9181.

%C The first counterexample to the conjecture is 6049681 which is 781 mod 990. 341 and 671 mod 990 also appear. Conjecture 1 is true; the only excluded possibility is 161 mod 210 which is never a 2-pseudoprime (check mod 7). All three parts of conjecture 2 are false; counterexamples include 461502097, 1378483393, and 7044493729. Conjecture 3 is true, consider Korselt's criterion mod 30. - _Charles R Greathouse IV_, Dec 07 2014

%H Charles R Greathouse IV, <a href="/A182090/b182090.txt">Table of n, a(n) for n = 1..10000</a>

%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>

%t CarmichaelNbrQ[n_] := ! PrimeQ[n] && Mod[n, CarmichaelLambda@ n] == 1; Select[ Range[10000000], CarmichaelNbrQ[ 11#] &] (* _Robert G. Wilson v_, Aug 24 2012 *)

%o (PARI) Korselt(n)=my(f=factor(n));for(i=1,#f~,if(f[i,2]>1||(n-1)%(f[i,1]-1), return(0)));1

%o is(n)=n%110==11 && Korselt(n) \\ _Charles R Greathouse IV_, Dec 07 2014

%Y Cf. A002997 (Carmichael numbers).

%K nonn

%O 1,1

%A _Marius Coman_, Apr 11 2012

%E Corrected by _Robert G. Wilson v_, Aug 24 2012