A356086 Intersection of A001952 and A022838.

3, 6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, 81, 88, 95, 102, 105, 109, 112, 116, 119, 122, 126, 129, 133, 136, 143, 150, 157, 174, 180, 187, 204, 211, 218, 221, 225, 228, 232, 235, 242, 245, 249, 252, 256, 259, 266, 273, 284, 285, 287, 289, 290, 292, 294

Offset: 1

Author

Clark Kimberling, Aug 04 2022

Keywords

• nonn
• easy

Comments

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

Examples

			(1)  u ^ v   = ( 1,  5,  8, 12, 15, 19, 22, 24, 25, 29, 31, 32, ...) = A346308
(2)  u ^ v'  = ( 2,  4,  7,  9, 11, 14, 16, 18, 21, 26, 28, 33, ...) = A356085
(3)  u' ^ v  = ( 3,  6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, ...) = A356086
(4)  u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087
		

Crossrefs

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

Programs

Mathematica

z = 200;
r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
r1 = Sqrt[3]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022838 *)
v1 = Take[Complement[Range[1000], v], z]  (* A054406 *)
Intersection[u, v]    (* A346308 *)
Intersection[u, v1]   (* A356085 *)
Intersection[u1, v]   (* A356086 *)
Intersection[u1, v1]  (* A356087 *)

Python

from math import isqrt
from itertools import count, islice
def A356086_gen(): # generator of terms
    return filter(lambda n:n == isqrt(3*(isqrt(n**2//3)+1)**2),((k:=n<<1)+isqrt(k*n) for n in count(1)))
A356086_list = list(islice(A356086_gen(),30)) # Chai Wah Wu, Aug 06 2022