A199824 Primes in successive intervals (2^i +1 .. 2^(i+1) -1) i=1,2,3,... such that there are no prime symmetric XOR couples in either the original interval or any recursively halved interval that contains them.

67, 167, 587, 719, 751, 769, 1129, 1163, 1531, 1913, 2099, 2153, 2543, 2819, 3049, 3079, 3709, 3967, 4691, 4861, 4909, 5147, 5347, 5749, 5813, 5939, 6121, 6151, 6397, 6473, 6563, 6709, 6883, 6899, 6911, 7247, 7393, 7451, 7703, 7829, 7919, 8093, 8171, 8447, 8707, 8807, 8963, 9157, 9161, 9209

Offset: 1

Author

Brad Clardy, Nov 11 2011

Keywords

• nonn

Comments

The MAGMA program provided produces output with each interval delimited by the power of 2 that starts it.

All of these primes are a sparse subset of isolated primes (the only possible exception would be a twin prime that crosses the interval boundary, but none are known to occur).

In each interval XOR couples are produced by XORing a number in the interval with 2^i -2 where i is the index used in the interval definition. In recursively halved intervals, i is decremented each time down to i=2.

Examples

			In the interval (17 .. 31) i=4 the numbers are coupled symmetrically around the middle of the interval by XORing each with 2^i -2 where i=4 or 14.
|-------XOR 14-------|
|  |--------------|  |
|  |  |--------|  |  |
|  |  |  |--|  |  |  |
17 19 21 23 25 27 29 31
(17,31), (19,29) are prime XOR couples but the prime 23 has a composite couple (23,25).
23 is in the first half of the interval. XORing each number in the first half of the interval with 2^i -2 where i=3 or 6
|---XOR 6---|
|   |---|   |
17  19  21  23
(17,23) is a prime XOR couple and all primes in the interval have been coupled, therefore there are no primes with only composite couples in the interval (17 .. 31).
The first such prime occurs in the interval (65 ..127) and is 67
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040.

Programs

Magma

XOR := func;
for i:= 4 to 16 do
    "****", i;
    for j:= 2^(i) +1 to 2^(i+1) -1 by 2 do
        sympair:=0;
        for k:= 2 to i do
            xornum:=2^k -2;
            xorcouple:=XOR(j,xornum);
            if (IsPrime(j) and IsPrime(xorcouple)) then sympair:=1;
               end if;
        end for;
        if ((sympair eq 0) and IsPrime(j)) then j;
           end if;
    end for;
end for;

Maple

q:= (l, p, r)-> r-l=2 or not isprime(l+r-p) and
                `if`((l+r)/2>p, q(l, p, (l+r)/2), q((l+r)/2, p, r)):
a:= proc(n) local p, l;
      p:= `if`(n=1, 3, a(n-1));
      do p:= nextprime(p);
         l:= 2^ilog2(p);
         if q(l, p, l+l) then break fi
      od; a(n):=p
    end:
seq(a(n), n=1..60); # Alois P. Heinz, Nov 13 2011

Mathematica

q[l_, p_, r_] := r - l == 2 || ! PrimeQ[l + r - p] &&
    If[(l + r)/2 > p, q[l, p, (l + r)/2], q[(l + r)/2, p, r]];
a[n_] := a[n] = Module[{p, l},
    p = If[n == 1, 3, a[n - 1]]; While[True, p = NextPrime[p];
    l = 2^(Length[IntegerDigits[p, 2]]-1); If[q[l, p, l+l], Break[]]]; p];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 11 2021, after Alois P. Heinz *)