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%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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