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A247607 Primes whose number of symmetric connections to other primes within the same power of two interval exceeds the number of symmetric connections to composites.

%I A247607 #26 Sep 08 2022 08:46:09
%S A247607 5,7,17,19,29,31,43,59,103,281,457,461,463,499,607,1409,1451,2143,
%T A247607 2657,4229,16063,19583,19699,62143,65537,70919,107347,113159,124783,
%U A247607 124981,600703,3103423,18936719
%N A247607 Primes whose number of symmetric connections to other primes within the same power of two interval exceeds the number of symmetric connections to composites.
%C A247607 While there may be some additional terms, it is thought to be a finite sequence. An exhaustive search was conducted up to 2^29.
%C A247607 Members larger than 2^8 will also be in A200321.
%e A247607 In the interval (2^4,2^5) [17,19,21,23,25,27,29,31], the prime 17 symmetrically couples with 31 around the midpoint of the interval, 23 around the midpoint of the halved interval, and 19 in the quartered interval. There is no composite couple. It has 3 prime and 0 composite symmetric connections. The same process for 19 produces 2 prime and 1 composite connection. Therefore 17 and 19 are members. 23 couples with 25, 17 and 22. It has 1 prime and 2 composite connections so it is not a member.
%o A247607 (Magma)
%o A247607 XOR := func<a, b | Seqint([ (adigs[i] + bdigs[i]) mod 2 : i in [1..n]], 2)
%o A247607   where adigs := Intseq(a, 2, n)
%o A247607   where bdigs := Intseq(b, 2, n)
%o A247607   where n := 1 + Ilog2(Max([a, b, 1]))>;
%o A247607 function PCcoord(X,i,P,C)
%o A247607 if (i eq 1) then
%o A247607     if (P gt C) then return true;
%o A247607       else
%o A247607       return false,P,C;
%o A247607     end if;
%o A247607   else
%o A247607     xornum:=2^i - 2;
%o A247607     xorcouple:=XOR(X, xornum);
%o A247607     if (IsPrime(xorcouple)) then
%o A247607        return PCcoord(X, i-1,P+1,C);
%o A247607     else
%o A247607        return PCcoord(X, i-1,P,C+1);
%o A247607     end if;
%o A247607   end if;
%o A247607 end function;
%o A247607 for k:= 1 to 2^10 + 1 by 2 do
%o A247607   if IsPrime(k) then
%o A247607      if PCcoord(k,Ilog2(k),0,0) then k;
%o A247607      end if;
%o A247607   end if;
%o A247607 end for;
%Y A247607 Cf. A200231.
%K A247607 nonn
%O A247607 1,1
%A A247607 _Brad Clardy_, Sep 22 2014