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

4,1

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.

Links

Table of n, a(n) for n=4..43.

Example

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;

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

Crossrefs

Cf. A200143.

Sequence in context: A123592 A260553 A165285 * A165981 A109998 A328998

Adjacent sequences: A200318 A200319 A200320 * A200322 A200323 A200324

Keyword

nonn,tabf

Author

Brad Clardy, Nov 15 2011

Extensions

More terms from Alois P. Heinz, Nov 16 2011

Status

approved