login
A360400
a(n) = A356133(A360392(n)).
4
7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, 83, 85, 89, 94, 97, 104, 110, 112, 115, 122, 127, 131, 137, 140, 142, 148, 155, 157, 161, 166, 169, 176, 182, 184, 187, 193, 200, 202, 208, 211, 215, 220, 223, 229, 236, 238, 244, 247, 251, 257
OFFSET
1,1
COMMENTS
This is the third 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, A286354, A286356, A360394 (intersections instead of results of composition), A360402-A360405.
Sequence in context: A243962 A287304 A023984 * A308200 A126621 A109331
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 11 2023
STATUS
approved