A182108
Odd composite numbers in successive intervals [2^i +1 .. 2^(i+1) -1] i=1,2,3... such that there are only composite symmetric XOR couples in either the original interval or any recursively halved interval that contains them.
Original entry on oeis.org
513, 695, 925, 1177, 1355, 1395, 1507, 1681, 1685, 1687, 1689, 1819, 1827, 1893, 1959, 2043, 2165, 2169, 2637, 2651, 2757, 2875, 2987, 3159, 3339, 3417, 3503, 3649, 3681, 3743, 3873, 3963, 3975, 4041, 4169, 4353, 4489, 4767, 4773, 4805, 4845, 4881, 5123
Offset: 1
-
XOR := func;
function IsClardynum(X,i)
if i eq 1 then
return true;
else
xornum:=2^i - 2;
xorcouple:=XOR(X,xornum);
if (IsPrime(xorcouple)) then
return false;
else
return IsClardynum(X,i-1);
end if;
end if;
end function;
for i:= 3 to 10001 by 2 do
if not IsPrime(i) then
if IsClardynum(i,Ilog2(i)) then i;
end if;
end if;
end for;
A200143
Nodes of degree 1 in graphs of XOR connected primes in successive intervals [2^i+1,2^(i+1)-1], i>=1.
Original entry on oeis.org
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
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.
-
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
-
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 *)
Showing 1-2 of 2 results.
Comments