cp's OEIS Frontend

A346308 Intersection of Beatty sequences for sqrt(2) and sqrt(3).

%I A346308 #22 Aug 06 2022 19:06:30
%S A346308 1,5,8,12,15,19,22,24,25,29,31,32,36,38,39,41,43,45,46,48,50,53,55,57,
%T A346308 60,62,65,67,69,72,74,76,77,79,83,84,86,90,91,93,96,98,100,103,107,
%U A346308 110,114,117,121,124,128,131,135,138,140,142,145,147,148,152,154
%N A346308 Intersection of Beatty sequences for sqrt(2) and sqrt(3).
%C A346308 Let d(n) = a(n) - A022840(n). Conjecture: (d(n)) is unbounded below and above, and d(n) = 0 for infinitely many n.
%C A346308 From _Clark Kimberling_, Jul 26 2022: (Start)
%C A346308 This is the first of four sequences that partition the positive integers. 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:
%C A346308 (1)  u ^ v = intersection of u and v (in increasing order);
%C A346308 (2)  u ^ v';
%C A346308 (3)  u' ^ v;
%C A346308 (4)  u' ^ v'.
%C A346308 Every positive integer is in exactly one of the four sequences. For A346308, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3)), so that r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2. (See A356052.)
%C A346308 (End)
%F A346308 In general, if r and s are irrational numbers greater than 1, and a(n) is the n-th term of the intersection (assumed nonempty) of the Beatty sequences for r and s, then a(n) = floor(r*ceiling(a(n)/r)) = floor(s*ceiling(a(n)/s)).
%e A346308 Beatty sequence for sqrt(2): (1,2,4,5,7,8,9,11,12,14,...).
%e A346308 Beatty sequence for sqrt(3): (1,3,5,6,8,10,12,13,15,...).
%e A346308 a(n) = (1,5,8,12,...).
%e A346308 In the notation in Comments:
%e A346308 (1)  u ^ v = (1, 5, 8, 12, 15, 19, 22, 24, 25, 29, 31, 32, ...) =  A346308.
%e A346308 (2)  u ^ v' = (2, 4, 7, 9, 11, 14, 16, 18, 21, 26, 28, 33, 35, ...) =  A356085.
%e A346308 (3)  u' ^ v = (3, 6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, 81, ...) = A356086.
%e A346308 (4)  u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087.
%t A346308 z = 200;
%t A346308 r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
%t A346308 u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
%t A346308 r1 = Sqrt[3]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022838 *)
%t A346308 v1 = Take[Complement[Range[1000], v], z]  (* A054406 *)
%t A346308 t1 = Intersection[u, v]    (* A346308 *)
%t A346308 t2 = Intersection[u, v1]   (* A356085 *)
%t A346308 t3 = Intersection[u1, v]   (* A356086 *)
%t A346308 t4 = Intersection[u1, v1]  (* A356087 *)
%o A346308 (Python)
%o A346308 from math import isqrt
%o A346308 from itertools import count, islice
%o A346308 def A346308_gen(): # generator of terms
%o A346308     return filter(lambda n:n == isqrt(3*(isqrt(n**2//3)+1)**2),(isqrt(n*n<<1) for n in count(1)))
%o A346308 A346308_list = list(islice(A346308_gen(),30)) # _Chai Wah Wu_, Aug 06 2022
%Y A346308 Intersection of A001951 and A022838.
%Y A346308 Cf. A346308, A347467, A347468, A347469.
%Y A346308 Cf. A001952, A022838, A054406, A356085, A356086, A356087, A356088 (composites instead of intersections).
%K A346308 nonn
%O A346308 1,2
%A A346308 _Clark Kimberling_, Sep 11 2021