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A200143 Nodes of degree 1 in graphs of XOR connected primes in successive intervals [2^i+1,2^(i+1)-1], i>=1.

%I A200143 #27 Nov 12 2020 06:29:40
%S A200143 5,7,11,13,23,47,61,83,131,191,211,223,241,317,331,397,467,479,491,
%T A200143 503,509,563,577,613,727,743,757,829,887,907,941,947,997,1009,1021,
%U A200143 1039,1069,1087,1097,1109,1223,1229,1237,1381,1399,1423,1447,1523,1543,1549
%N A200143 Nodes of degree 1 in graphs of XOR connected primes in successive intervals [2^i+1,2^(i+1)-1], i>=1.
%C A200143 The number used to produce the XOR couple is 2^i-2, with i sharing the index value of the initial interval and decremented in halved intervals down to 2.
%H A200143 Alois P. Heinz, <a href="/A200143/b200143.txt">Table of n, a(n) for n = 1..10000</a>
%e A200143 In the interval [17,31], i=4, the XOR couple number is 2^4-2=14. For half intervals it is 2^3-2 = 6, and the final application would be 2^2-2 = 2. All of the pairings can be represented as:
%e A200143 |-------XOR 14-------|
%e A200143 |  |--------------|  |
%e A200143 |  |  |--------|  |  |
%e A200143 |  |  |  |--|  |  |  |
%e A200143 17 19 21 23 25 27 29 31
%e A200143 |-XOR  6-|  |-XOR  6-|
%e A200143 |  |--|  |  |  |--|  |
%e A200143 17 19 21 23 25 27 29 31
%e A200143 XOR   XOR   XOR   XOR
%e A200143 |2-|  |2-|  |2-|  |2-|
%e A200143 17 19 21 23 25 27 29 31
%e A200143 The prime XOR couples are (17,31), (19,29), (17,23), (17,19), (29,31) which produces the graph:
%e A200143   17 19 23 29 31
%e A200143 17 0  1  1  0  1             19
%e A200143 19 1  0  0  1  0           /    \
%e A200143 23 1  0  0  0  0   or  23~17~31~29
%e A200143 29 0  1  0  0  1
%e A200143 31 1  0  0  1  0
%e A200143 Therefore 23 is the only node of degree 1 in the interval.
%p A200143 q:= (l, p, r)-> `if`(r-l=2, 0, `if`(isprime(l+r-p), 1, 0)+
%p A200143                 `if`((l+r)/2>p, q(l, p, (l+r)/2), q((l+r)/2, p, r))):
%p A200143 a:= proc(n) local p, l;
%p A200143       p:= `if`(n=1, 1, a(n-1));
%p A200143       do p:= nextprime(p);
%p A200143          l:= 2^ilog2(p);
%p A200143          if q(l, p, l+l)=1 then break fi
%p A200143       od; a(n):=p
%p A200143     end:
%p A200143 seq(a(n), n=1..60);  # _Alois P. Heinz_, Nov 15 2011
%t A200143 q[l_, p_, r_] := q[l, p, r] = If[r - l == 2, 0, If[PrimeQ[l + r - p], 1, 0] + If[(l + r)/2 > p, q[l, p, (l + r)/2], q[(l + r)/2, p, r]]];
%t A200143 a[n_] := a[n] = Module[{p, l}, p = If[n==1, 1, a[n-1]]; While[True, p = NextPrime[p]; l = 2^Floor@Log[2, p]; If[q[l, p, l+l]==1, Break[]]]; p];
%t A200143 Array[a, 60] (* _Jean-François Alcover_, Nov 12 2020, after _Alois P. Heinz_ *)
%Y A200143 Cf. A199824.
%K A200143 nonn
%O A200143 1,1
%A A200143 _Brad Clardy_, Nov 14 2011