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.

5, 7, 17, 19, 29, 31, 43, 59, 103, 281, 457, 461, 463, 499, 607, 1409, 1451, 2143, 2657, 4229, 16063, 19583, 19699, 62143, 65537, 70919, 107347, 113159, 124783, 124981, 600703, 3103423, 18936719

Offset

1,1

Comments

While there may be some additional terms, it is thought to be a finite sequence. An exhaustive search was conducted up to 2^29.

Members larger than 2^8 will also be in A200321.

Links

Table of n, a(n) for n=1..33.

Example

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.

Prog

(Magma)
XOR := func<a, b | Seqint([ (adigs[i] + bdigs[i]) mod 2 : i in [1..n]], 2)
  where adigs := Intseq(a, 2, n)
  where bdigs := Intseq(b, 2, n)
  where n := 1 + Ilog2(Max([a, b, 1]))>;
function PCcoord(X, i, P, C)
if (i eq 1) then
    if (P gt C) then return true;
      else
      return false, P, C;
    end if;
  else
    xornum:=2^i - 2;
    xorcouple:=XOR(X, xornum);
    if (IsPrime(xorcouple)) then
       return PCcoord(X, i-1, P+1, C);
    else
       return PCcoord(X, i-1, P, C+1);
    end if;
  end if;
end function;
for k:= 1 to 2^10 + 1 by 2 do
  if IsPrime(k) then
     if PCcoord(k, Ilog2(k), 0, 0) then k;
     end if;
  end if;
end for;

Crossrefs

Cf. A200231.

Sequence in context: A003630 A122565 A369105 * A079016 A106120 A106121

Adjacent sequences: A247604 A247605 A247606 * A247608 A247609 A247610

Keyword

nonn

Author

Brad Clardy, Sep 22 2014

Status

approved