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A356102 Intersection of A001950 and A022839.

%I A356102 #15 Nov 19 2022 20:35:15
%S A356102 2,13,15,20,26,31,44,49,60,62,73,78,89,91,96,102,107,109,120,125,136,
%T A356102 138,143,149,154,167,172,178,183,185,196,201,212,214,219,225,230,243,
%U A356102 248,259,261,272,277,290,295,301,306,308,319,324,326,328,330,333,335
%N A356102 Intersection of A001950 and A022839.
%C A356102 This is the third of four sequences that partition the positive integers. See A351415.
%e A356102 Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
%e A356102 (1)  u ^ v = intersection of u and v (in increasing order);
%e A356102 (2)  u ^ v';
%e A356102 (3)  u' ^ v;
%e A356102 (4)  u' ^ v'.
%e A356102 Every positive integer is in exactly one of the four sequences. For A351415, 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.
%e A356102 (1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
%e A356102 (2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
%e A356102 (3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
%e A356102 (4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103
%t A356102 z = 200;
%t A356102 r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
%t A356102 u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
%t A356102 r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
%t A356102 v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
%t A356102 Intersection[u, v]   (* A351415 *)
%t A356102 Intersection[u, v1]  (* A356101 *)
%t A356102 Intersection[u1, v]  (* A356102 *)
%t A356102 Intersection[u1, v1] (* A356103 *)
%Y A356102 Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356101, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).
%K A356102 nonn,easy
%O A356102 1,1
%A A356102 _Clark Kimberling_, Sep 04 2022