A356139 - OEIS

%I #11 Aug 24 2022 09:51:09

%S 2,4,8,10,14,16,18,23,25,29,31,33,37,39,43,46,50,52,54,58,60,64,67,69,

%T 73,75,79,81,85,87,90,94,96,100,102,104,108,110,115,117,119,123,125,

%U 129,131,136,138,140,144,146,150,152,154,159,161,165,167,171,173

%N a(n) = A137804(A001951(n)).

%C 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:

%C (1) v o u, defined by (v o u)(n)  = v(u(n));

%C (2) v' o u;

%C (3) v o u';

%C (4) v' o u'.

%C Every positive integer is in exactly one of the four sequences.

%C For the reverse composites, u o v, u o v', u' o v, u' o v', see A356056 to A356059.

%C 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

%C 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.

%C For A356139, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*(1/2 + sqrt(2)), so that r = sqrt(2), s = 1/2 + sqrt(2), r' = 2 + sqrt(2), s' = (9 + 4*sqrt(2))/7.

%e (1)  v o u   = (1,  3,  7,  9, 13, 15, 17, 21, 22, 26, 28, 30, 34, ...) = A356138

%e (2)  v' o u  = (2,  4,  8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, ...) = A356139

%e (3)  v o u'  = (5, 11, 19, 24, 32, 38, 44, 51, 57, 65, 70, 76, 84, ...) = A356140

%e (4)  v' o u' = (6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, ...) = A356141

%t z = 800;

%t u = Table[Floor[n (Sqrt[2])], {n, 1, z}];   (*A001951*)

%t u1 = Complement[Range[Max[u]], u] ;    (*A001952*)

%t v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}];  (*A137803*)

%t v1 = Complement[Range[Max[v]], v] ;     (*A137804*)

%t Table[v[[u[[n]]]], {n, 1, z/8}]   (*A356138 *)

%t Table[v1[[u[[n]]]], {n, 1, z/8}]  (* A356139*)

%t Table[v[[u1[[n]]]], {n, 1, z/8}]  (* A356140 *)

%t Table[v1[[u1[[n]]]], {n, 1, z/8}] (* A356141 *)

%Y Cf. A001951, A001952, A137804.

%Y Cf. A356056, A356057, A356058, A356059, A356138, A356140, A356141.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Aug 06 2022