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A182116 Carmichael numbers that only have composite XOR couples as defined in A182108.

%I A182116 #24 Sep 08 2022 08:45:54
%S A182116 410041,19384289,41341321,43620409,69331969,93030145,122785741,
%T A182116 130032865,133344793,133800661,157731841,238527745,334783585,
%U A182116 396262945,403043257,413631505,417241045,477726145,490503601,561777121,631071001,686059921,707926801,854197345
%N A182116 Carmichael numbers that only have composite XOR couples as defined in A182108.
%C A182116 There are 255 Carmichael numbers below 10^8 but only 6 of them have this property.
%H A182116 Jon E. Schoenfield, <a href="/A182116/b182116.txt">Table of n, a(n) for n = 1..427</a>
%o A182116 (Magma)
%o A182116 XOR := func<a, b | Seqint([ (adigs[i] + bdigs[i]) mod 2 : i in [1..n]], 2)
%o A182116     where adigs := Intseq(a, 2, n)
%o A182116     where bdigs := Intseq(b, 2, n)
%o A182116     where n := 1 + Ilog2(Max([a, b, 1]))>;
%o A182116 function IsClardynum(X,i)
%o A182116   if i eq 1 then
%o A182116     return true;
%o A182116   else
%o A182116     xornum:=2^i - 2;
%o A182116     xorcouple:=XOR(X,xornum);
%o A182116     if (IsPrime(xorcouple)) then
%o A182116        return false;
%o A182116     else
%o A182116        return IsClardynum(X,i-1);
%o A182116     end if;
%o A182116   end if;
%o A182116 end function;
%o A182116 function Korselt(X,n);
%o A182116 i:=1;
%o A182116 while IsDefined(X,i) do
%o A182116    b:=(n-1)mod(X[i]-1);
%o A182116    if (b ne 0) then return false;
%o A182116       else i:=i+1;
%o A182116    end if;
%o A182116 end while;
%o A182116 return true;
%o A182116 end function;
%o A182116 function IsCarmichael(n);
%o A182116   if IsPrime(n) then return false;
%o A182116   end if;
%o A182116   A:=AssociativeArray();
%o A182116   if IsSquarefree(n) then
%o A182116      A:=PrimeDivisors(n);
%o A182116      if Korselt(A,n) then return true;
%o A182116         else return false;
%o A182116      end if;
%o A182116      else
%o A182116      return false;
%o A182116   end if;
%o A182116 end function;
%o A182116 for i:=561 to 100000001 by 2 do
%o A182116     if IsCarmichael(i) then
%o A182116        if IsClardynum(i,Ilog2(i)) then i;
%o A182116        end if;
%o A182116     end if;
%o A182116 end for;
%Y A182116 Cf. A002997, A182108.
%K A182116 nonn
%O A182116 1,1
%A A182116 _Brad Clardy_, Apr 12 2012
%E A182116 a(11)-a(19) by _Brad Clardy_, May 10 2014
%E A182116 More terms and b-file (using the Magma program by _Brad Clardy_ and the b-file of Carmichael numbers from A002997) from _Jon E. Schoenfield_, May 10 2014