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A182116 Carmichael numbers that only have composite XOR couples as defined in A182108. 3

%I #24 Sep 08 2022 08:45:54

%S 410041,19384289,41341321,43620409,69331969,93030145,122785741,

%T 130032865,133344793,133800661,157731841,238527745,334783585,

%U 396262945,403043257,413631505,417241045,477726145,490503601,561777121,631071001,686059921,707926801,854197345

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

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

%H Jon E. Schoenfield, <a href="/A182116/b182116.txt">Table of n, a(n) for n = 1..427</a>

%o (Magma)

%o XOR := func<a, b | Seqint([ (adigs[i] + bdigs[i]) mod 2 : i in [1..n]], 2)

%o where adigs := Intseq(a, 2, n)

%o where bdigs := Intseq(b, 2, n)

%o where n := 1 + Ilog2(Max([a, b, 1]))>;

%o function IsClardynum(X,i)

%o if i eq 1 then

%o return true;

%o else

%o xornum:=2^i - 2;

%o xorcouple:=XOR(X,xornum);

%o if (IsPrime(xorcouple)) then

%o return false;

%o else

%o return IsClardynum(X,i-1);

%o end if;

%o end if;

%o end function;

%o function Korselt(X,n);

%o i:=1;

%o while IsDefined(X,i) do

%o b:=(n-1)mod(X[i]-1);

%o if (b ne 0) then return false;

%o else i:=i+1;

%o end if;

%o end while;

%o return true;

%o end function;

%o function IsCarmichael(n);

%o if IsPrime(n) then return false;

%o end if;

%o A:=AssociativeArray();

%o if IsSquarefree(n) then

%o A:=PrimeDivisors(n);

%o if Korselt(A,n) then return true;

%o else return false;

%o end if;

%o else

%o return false;

%o end if;

%o end function;

%o for i:=561 to 100000001 by 2 do

%o if IsCarmichael(i) then

%o if IsClardynum(i,Ilog2(i)) then i;

%o end if;

%o end if;

%o end for;

%Y Cf. A002997, A182108.

%K nonn

%O 1,1

%A _Brad Clardy_, Apr 12 2012

%E a(11)-a(19) by _Brad Clardy_, May 10 2014

%E 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

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Last modified February 25 02:17 EST 2024. Contains 370308 sequences. (Running on oeis4.)