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

410041, 19384289, 41341321, 43620409, 69331969, 93030145, 122785741, 130032865, 133344793, 133800661, 157731841, 238527745, 334783585, 396262945, 403043257, 413631505, 417241045, 477726145, 490503601, 561777121, 631071001, 686059921, 707926801, 854197345

Offset: 1

Author

Brad Clardy, Apr 12 2012

Keywords

• nonn

Comments

There are 255 Carmichael numbers below 10^8 but only 6 of them have this property.

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..427

Crossrefs

Cf. A002997, A182108.

Programs

Magma

XOR := func;
function IsClardynum(X,i)
  if i eq 1 then
    return true;
  else
    xornum:=2^i - 2;
    xorcouple:=XOR(X,xornum);
    if (IsPrime(xorcouple)) then
       return false;
    else
       return IsClardynum(X,i-1);
    end if;
  end if;
end function;
function Korselt(X,n);
i:=1;
while IsDefined(X,i) do
   b:=(n-1)mod(X[i]-1);
   if (b ne 0) then return false;
      else i:=i+1;
   end if;
end while;
return true;
end function;
function IsCarmichael(n);
  if IsPrime(n) then return false;
  end if;
  A:=AssociativeArray();
  if IsSquarefree(n) then
     A:=PrimeDivisors(n);
     if Korselt(A,n) then return true;
        else return false;
     end if;
     else
     return false;
  end if;
end function;
for i:=561 to 100000001 by 2 do
    if IsCarmichael(i) then
       if IsClardynum(i,Ilog2(i)) then i;
       end if;
    end if;
end for;

Extensions

a(11)-a(19) by Brad Clardy, May 10 2014

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