cp's OEIS Frontend

A356138 a(n) = A137803(A001951(n)).

%I A356138 #16 Mar 23 2025 18:38:28
%S A356138 1,3,7,9,13,15,17,21,22,26,28,30,34,36,40,42,45,47,49,53,55,59,61,63,
%T A356138 66,68,72,74,78,80,82,86,88,91,93,95,99,101,105,107,109,112,114,118,
%U A356138 120,124,126,128,132,133,137,139,141,145,147,151,153,156,158,160
%N A356138 a(n) = A137803(A001951(n)).
%C A356138 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:
%C A356138 (1) v o u, defined by (v o u)(n) = v(u(n));
%C A356138 (2) v' o u;
%C A356138 (3) v o u';
%C A356138 (4) v' o u'.
%C A356138 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 A356056 to A356059.
%C A356138 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 A356138 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
%C A356138 For A356138, 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 A356138 (1)  v o u   = (1,  3,  7,  9, 13, 15, 17, 21, 22, 26, 28, 30, 34, ...) = A356138
%e A356138 (2)  v' o u  = (2,  4,  8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, ...) = A356139
%e A356138 (3)  v o u'  = (5, 11, 19, 24, 32, 38, 44, 51, 57, 65, 70, 76, 84, ...) = A356140
%e A356138 (4)  v' o u' = (6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, ...) = A356141
%t A356138 z = 800;
%t A356138 u = Table[Floor[n (Sqrt[2])], {n, 1, z}];   (* A001951 *)
%t A356138 u1 = Complement[Range[Max[u]], u] ;    (* A001952 *)
%t A356138 v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}];  (* A137803 *)
%t A356138 v1 = Complement[Range[Max[v]], v] ;     (* A137804 *)
%t A356138 Table[v[[u[[n]]]], {n, 1, z/8}]   (* A356138 *)
%t A356138 Table[v1[[u[[n]]]], {n, 1, z/8}]  (* A356139 *)
%t A356138 Table[v[[u1[[n]]]], {n, 1, z/8}]  (* A356140 *)
%t A356138 Table[v1[[u1[[n]]]], {n, 1, z/8}] (* A356141 *)
%Y A356138 Cf. A001951, A001952, A137804.
%Y A356138 Cf. A356056, A356057, A356058, A356059, A356139, A356140, A356141.
%K A356138 nonn,easy
%O A356138 1,2
%A A356138 _Clark Kimberling_, Aug 06 2022