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Intersection of A001952 and A022838.
4

%I #11 Aug 06 2022 19:07:20

%S 3,6,10,13,17,20,27,34,51,58,64,71,81,88,95,102,105,109,112,116,119,

%T 122,126,129,133,136,143,150,157,174,180,187,204,211,218,221,225,228,

%U 232,235,242,245,249,252,256,259,266,273,284,285,287,289,290,292,294

%N Intersection of A001952 and A022838.

%C This is the third of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A346308.

%e (1) u ^ v = ( 1, 5, 8, 12, 15, 19, 22, 24, 25, 29, 31, 32, ...) = A346308

%e (2) u ^ v' = ( 2, 4, 7, 9, 11, 14, 16, 18, 21, 26, 28, 33, ...) = A356085

%e (3) u' ^ v = ( 3, 6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, ...) = A356086

%e (4) u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087

%t z = 200;

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

%t u1 = Take[Complement[Range[1000], u], z] (* A001952 *)

%t r1 = Sqrt[3]; v = Table[Floor[n*r1], {n, 1, z}] (* A022838 *)

%t v1 = Take[Complement[Range[1000], v], z] (* A054406 *)

%t Intersection[u, v] (* A346308 *)

%t Intersection[u, v1] (* A356085 *)

%t Intersection[u1, v] (* A356086 *)

%t Intersection[u1, v1] (* A356087 *)

%o (Python)

%o from math import isqrt

%o from itertools import count, islice

%o def A356086_gen(): # generator of terms

%o return filter(lambda n:n == isqrt(3*(isqrt(n**2//3)+1)**2),((k:=n<<1)+isqrt(k*n) for n in count(1)))

%o A356086_list = list(islice(A356086_gen(),30)) # _Chai Wah Wu_, Aug 06 2022

%Y Cf. A001951, A001952, A022838, A054406, A346308, A356085, A356087, A356088 (results of composition instead of intersections).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Aug 04 2022