A200321 Irregular triangle T(n,k) where row n contains the maximal nodes in the graph of XOR connected primes of interval [2^n+1,2^(n+1)-1], n>=4.

17, 43, 59, 103, 139, 151, 157, 173, 193, 281, 457, 461, 463, 499, 607, 1409, 1451, 2143, 2657, 4229, 16063, 19583, 19699, 62143, 124981, 166303, 172663, 240257, 244301, 276041, 289853, 305411, 327319, 376639, 417941, 505663, 518761, 524119, 600703, 1056287

Offset: 4

Author

Brad Clardy, Nov 15 2011

Keywords

• nonn
• tabf

Comments

Nodes with degree > 2 that have the greatest number of vertices in prime XOR connected graphs are defined as maximal nodes. The graph is constructed in the manner outlined in A200143.

Examples

			The XOR connected graph for the interval [33,63], n=5, is
   37 41 43 47 53 59 61
37  0  0  1  0  0  1  0
41  0  0  1  1  0  0  0             37
43  1  1  0  0  1  0  0            /  \
47  0  1  0  0  0  0  0 or 47~41~43   59~61
53  0  0  1  0  0  1  0            \  /
59  1  0  0  0  1  0  1             53
61  0  0  0  0  0  1  0
The maximum number of vertices connected to any prime is 3, therefore 43 and 59 are members of row n=5.
Triangle begins:
17;
43,   59;
103;
139,  151,  157,  173,  193;
281,  457,  461,  463,  499;
607;
1409, 1451;
		

Crossrefs

Cf. A200143.

Programs

Maple

q:= (l, p, r)-> `if`(r-l=2, 0, `if`(isprime(l+r-p), 1, 0)+
                `if`((l+r)/2>p, q(l, p, (l+r)/2), q((l+r)/2, p, r))):
T:= proc(n) local r, l, u, p, m, d;
      r:= NULL;
      l:= 2^n; u:= 2*l;
      p:= nextprime(l);
      m:= -1;
      while p<=u do
        d:= q(l, p, u);
        if d=m then r:= r,p
      elif d>m then m:= d; r:= p fi;
        p:= nextprime(p)
      od;
      `if`(m>=3, r, NULL)
    end:
seq(T(n), n=4..18);  # Alois P. Heinz, Nov 16 2011

Extensions

More terms from Alois P. Heinz, Nov 16 2011