



2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, 49, 52, 56, 59, 61, 66, 68, 73, 75, 78, 82, 85, 89, 92, 97, 99, 101, 106, 108, 113, 115, 118, 123, 125, 130, 132, 134, 139, 141, 146, 149, 153, 156, 158, 163, 165, 170, 172, 175, 179, 182, 186, 189, 194
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OFFSET

1,1


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) v o u, defined by (v o u)(n) = v(u(n));
(2) v' o u;
(3) v o u';
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. For the reverse composites, u o v, u o v', u' o v, u' o v', see A356088 to A356091.
Assume that if w is any of the sequences u, v, u', v', then lim_{n>oo) w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356181, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3)), so that r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2.


LINKS



EXAMPLE

(1) v o u = (1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, ...) = A356180
(2) v' o u = (2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, ...) = A356181
(3) v o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, ...) = A356182
(4) v' o u' = (7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, ...) = A356183


MATHEMATICA

z = 800; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}]; (* A001951 *)
u1 = Complement[Range[Max[u]], u]; (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}]; (* A022838 *)
v1 = Complement[Range[Max[v]], v]; (* A054406 *)
Table[v[[u[[n]]]], {n, 1, zz}] (* A356180 *)
Table[v1[[u[[n]]]], {n, 1, zz}] (* A356181) *)
Table[v[[u1[[n]]]], {n, 1, zz}] (* A356182 *)
Table[v1[[u1[[n]]]], {n, 1, zz}] (* A356183 *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



