cp's OEIS Frontend

A356103 Intersection of A001950 and A108598.

%I A356103 #15 Nov 19 2022 20:35:26
%S A356103 5,7,10,18,23,28,34,36,39,41,47,52,54,57,65,68,70,75,81,83,86,94,99,
%T A356103 104,112,115,117,123,128,130,133,141,146,151,157,159,162,164,170,175,
%U A356103 180,188,191,193,198,204,206,209,217,222,227,233,235,238,240,246,251
%N A356103 Intersection of A001950 and A108598.
%C A356103 This is the fourth of four sequences that partition the positive integers. See A351415.
%e A356103 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 A356103 (1)  u ^ v = intersection of u and v (in increasing order);
%e A356103 (2)  u ^ v';
%e A356103 (3)  u' ^ v;
%e A356103 (4)  u' ^ v'.
%e A356103 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 A356103 (1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
%e A356103 (2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
%e A356103 (3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
%e A356103 (4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103
%t A356103 z = 200;
%t A356103 r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
%t A356103 u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
%t A356103 r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
%t A356103 v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
%t A356103 Intersection[u, v]   (* A351415 *)
%t A356103 Intersection[u, v1]  (* A356101 *)
%t A356103 Intersection[u1, v]  (* A356102 *)
%t A356103 Intersection[u1, v1] (* A356103 *)
%Y A356103 Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356101, A356102, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).
%K A356103 nonn,easy
%O A356103 1,1
%A A356103 _Clark Kimberling_, Sep 04 2022