



3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, 50, 53, 56, 61, 64, 67, 71, 74, 79, 82, 85, 88, 93, 97, 100, 103, 108, 111, 114, 118, 122, 126, 129, 132, 135, 140, 144, 147, 150, 155, 158, 161, 165, 169, 173, 176, 179, 184, 187, 190, 194, 197, 202, 205, 208
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OFFSET

1,1


COMMENTS

This is the first 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 u';
(3) u' o v;
(4) u' o v'.
Every positive integer is in exactly one of the four sequences. For the reverse composites, v o u, v' o u, v o u', v' o u', see A356217 to A356220.
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 A356104, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.


LINKS

Table of n, a(n) for n=1..58.


EXAMPLE

(1) u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2) u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3) u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = A356106
(4) u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107


MATHEMATICA

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}]; (* A000201 *)
u1 = Complement[Range[Max[u]], u]; (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}]; (* A022839 *)
v1 = Complement[Range[Max[v]], v]; (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}] (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}] (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}] (* A356106 *)
Table[u1[[v1[[n]]]], {n, 1, zz}] (* A356107 *)


CROSSREFS

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356105, A356106, A356107, A351415 (intersections), A356217 (reverse composites).
Sequence in context: A061796 A113241 A310153 * A127621 A049707 A213685
Adjacent sequences: A356101 A356102 A356103 * A356105 A356106 A356107


KEYWORD

nonn,easy,changed


AUTHOR

Clark Kimberling, Sep 08 2022


STATUS

approved



