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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.
2
5, 7, 11, 13, 23, 47, 61, 83, 131, 191, 211, 223, 241, 317, 331, 397, 467, 479, 491, 503, 509, 563, 577, 613, 727, 743, 757, 829, 887, 907, 941, 947, 997, 1009, 1021, 1039, 1069, 1087, 1097, 1109, 1223, 1229, 1237, 1381, 1399, 1423, 1447, 1523, 1543, 1549
OFFSET
1,1
COMMENTS
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.
LINKS
EXAMPLE
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:
|-------XOR 14-------|
| |--------------| |
| | |--------| | |
| | | |--| | | |
17 19 21 23 25 27 29 31
|-XOR 6-| |-XOR 6-|
| |--| | | |--| |
17 19 21 23 25 27 29 31
XOR XOR XOR XOR
|2-| |2-| |2-| |2-|
17 19 21 23 25 27 29 31
The prime XOR couples are (17,31), (19,29), (17,23), (17,19), (29,31) which produces the graph:
17 19 23 29 31
17 0 1 1 0 1 19
19 1 0 0 1 0 / \
23 1 0 0 0 0 or 23~17~31~29
29 0 1 0 0 1
31 1 0 0 1 0
Therefore 23 is the only node of degree 1 in the interval.
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))):
a:= proc(n) local p, l;
p:= `if`(n=1, 1, a(n-1));
do p:= nextprime(p);
l:= 2^ilog2(p);
if q(l, p, l+l)=1 then break fi
od; a(n):=p
end:
seq(a(n), n=1..60); # Alois P. Heinz, Nov 15 2011
MATHEMATICA
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]]];
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];
Array[a, 60] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz *)
CROSSREFS
Cf. A199824.
Sequence in context: A022885 A176831 A263467 * A265780 A135774 A180946
KEYWORD
nonn
AUTHOR
Brad Clardy, Nov 14 2011
STATUS
approved