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A360399
a(n) = A026430(1 + A360393(n)).
4
1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, 73, 77, 82, 87, 90, 96, 99, 103, 109, 114, 117, 121, 128, 130, 136, 141, 144, 149, 154, 159, 162, 168, 171, 175, 181, 186, 189, 194, 199, 203, 209, 213, 216, 222, 225, 230, 235, 239, 245, 249
OFFSET
1,2
COMMENTS
This is the second of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).
EXAMPLE
(1) u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2) u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3) u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4) u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401
MATHEMATICA
z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u]; (* A356133 *)
v = u + 2; (* A360392 *)
v1 = Complement[Range[Max[v]], v]; (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}] (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}] (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}] (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}] (* A360401 *)
CROSSREFS
Cf. A026530, A356133, A360392, A360393, A360398, A286355, A286356, A360394 (intersections instead of results of composition), A360402-A360405.
Sequence in context: A310165 A310166 A310167 * A310168 A134031 A228172
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 10 2023
STATUS
approved