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A360397
Intersection of A356133 and A360393.
8
2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, 112, 124, 133, 142, 148, 157, 166, 178, 184, 193, 202, 208, 220, 229, 238, 244, 253, 262, 274, 280, 292, 301, 310, 322, 328, 337, 346, 352, 364, 373, 382, 394, 400, 412, 421, 430, 436, 445, 454, 466, 472
OFFSET
1,1
COMMENTS
This is the fourth of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1) u ^ v = intersection of u and v (in increasing order);
(2) u ^ v';
(3) u' ^ v;
(4) u' ^ v'.
Every positive integer is in exactly one of the four sequences. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively.
For A360397, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430;
u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u;
v = u + 1 = A285954, except its initial 1;
v' = complement of v.
EXAMPLE
(1) u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) = A360394
(2) u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) = A360395
(3) u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4) u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397
MATHEMATICA
z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u]; (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v]; (* A360393 *)
Intersection[u, v] (* A360394 *)
Intersection[u, v1] (* A360395 *)
Intersection[u1, v] (* A360396 *)
Intersection[u1, v1] (* A360397 *)
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 10 2023
STATUS
approved