A182108 -id:A182108 - cp's OEIS frontend

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

Brad Clardy, Apr 12 2012

nonn

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

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

Cf. A002997, A182108.

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;

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

A252944 Fermat pseudoprimes that are not Carmichael numbers and have only composite XOR couples as defined in A182108.

Original entry on oeis.org

23377, 31417, 49981, 74665, 220729, 435671, 679729, 769757, 852481, 915981, 1016801, 1023121, 1128121, 1397419, 2008597, 2987167, 3073357, 4014361

Offset: 1

Brad Clardy, Dec 25 2014

There are 433 Fermat pseudoprimes that aren't Carmichael numbers below 2^22, but only 18 have this property. Carmichael numbers that have this property are in A182116.

Cf. A153508, A001567, A252943, A182108, A182116.

function IsClardynum(X, i)
  if i eq 1 then
    return true;
  else
    xornum:=2^i - 2;
    xorcouple:=BitwiseXor(X, xornum);
    if (IsPrime(xorcouple)) then
       return false;
    else
       return IsClardynum(X, i-1);
    end if;
  end if;
end function;
for n:= 3 to 1052503 by 2 do
  if (IsOne(2^(n-1) mod n)
      and not IsPrime(n)
      and not n mod CarmichaelLambda(n) eq 1
      and IsClardynum(n,Ilog2(n)))
      then n;
  end if;
end for;

A242435 Number of terms of A182116 between 2^n and 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 6, 2, 7, 8, 6, 12, 11, 21, 24, 27, 35, 45, 68, 86, 117, 176, 206, 260, 370, 457, 565, 750, 967, 1321, 1531, 1978, 2842, 3723, 4587, 5677, 8354, 10708, 13435, 17259, 23040, 31741, 40146, 48596, 66728, 92193, 112771, 149002, 209890

Offset: 1

Brad Clardy, May 14 2014

nonn

This was done with data on Carmichael numbers below 10^21 provided by R. G. E. Pinch, and special computational assistance from William Stein.

There are 16396564 Carmichael numbers below 2^69 but only 849752 have the property of A182116. It looks as though the ratio of Carmichael numbers of this type to normal Carmichael numbers converges to a value around 0.051.

Richard Pinch, Carmichael numbers up to 10^21

Cf. A002997, A182116, A182108, A182490.

cp's OEIS Frontend

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

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A252944 Fermat pseudoprimes that are not Carmichael numbers and have only composite XOR couples as defined in A182108.

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Author

Keywords

Comments

Crossrefs

Programs

Magma

A242435 Number of terms of A182116 between 2^n and 2^(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs