cp's OEIS Frontend

A182108 Odd composite numbers in successive intervals [2^i +1 .. 2^(i+1) -1] i=1,2,3... such that there are only composite symmetric XOR couples in either the original interval or any recursively halved interval that contains them.

%I A182108 #7 Sep 08 2022 08:45:54
%S A182108 513,695,925,1177,1355,1395,1507,1681,1685,1687,1689,1819,1827,1893,
%T A182108 1959,2043,2165,2169,2637,2651,2757,2875,2987,3159,3339,3417,3503,
%U A182108 3649,3681,3743,3873,3963,3975,4041,4169,4353,4489,4767,4773,4805,4845,4881,5123
%N A182108 Odd composite numbers in successive intervals [2^i +1 .. 2^(i+1) -1] i=1,2,3... such that there are only composite symmetric XOR couples in either the original interval or any recursively halved interval that contains them.
%C A182108 The description of the process is outlined in A199824. Up to the interval that starts 2^10 there are only 109 of these numbers, while there are a mere 50 primes of the type in A199824.
%o A182108 (Magma)
%o A182108 XOR := func<a, b | Seqint([ (adigs[i] + bdigs[i]) mod 2 : i in [1..n]], 2)
%o A182108     where adigs := Intseq(a, 2, n)
%o A182108     where bdigs := Intseq(b, 2, n)
%o A182108     where n := 1 + Ilog2(Max([a, b, 1]))>;
%o A182108 function IsClardynum(X,i)
%o A182108   if i eq 1 then
%o A182108     return true;
%o A182108   else
%o A182108     xornum:=2^i - 2;
%o A182108     xorcouple:=XOR(X,xornum);
%o A182108     if (IsPrime(xorcouple)) then
%o A182108        return false;
%o A182108     else
%o A182108        return IsClardynum(X,i-1);
%o A182108     end if;
%o A182108   end if;
%o A182108 end function;
%o A182108 for i:= 3 to 10001 by 2 do
%o A182108    if not IsPrime(i) then
%o A182108       if IsClardynum(i,Ilog2(i)) then i;
%o A182108       end if;
%o A182108    end if;
%o A182108 end for;
%Y A182108 Cf. A199824.
%K A182108 nonn
%O A182108 1,1
%A A182108 _Brad Clardy_, Apr 12 2012