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 A346308 Intersection of Beatty sequences for sqrt(2) and sqrt(3). 17
 1, 5, 8, 12, 15, 19, 22, 24, 25, 29, 31, 32, 36, 38, 39, 41, 43, 45, 46, 48, 50, 53, 55, 57, 60, 62, 65, 67, 69, 72, 74, 76, 77, 79, 83, 84, 86, 90, 91, 93, 96, 98, 100, 103, 107, 110, 114, 117, 121, 124, 128, 131, 135, 138, 140, 142, 145, 147, 148, 152, 154 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Let d(n) = a(n) - A022840(n). Conjecture: (d(n)) is unbounded below and above, and d(n) = 0 for infinitely many n. From Clark Kimberling, Jul 26 2022: (Start) 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: (1) u ^ v = intersection of u and v (in increasing order); (2) u ^ v'; (3) u' ^ v; (4) u' ^ v'. 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.) (End) LINKS Table of n, a(n) for n=1..61. FORMULA 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)). EXAMPLE Beatty sequence for sqrt(2): (1,2,4,5,7,8,9,11,12,14,...). Beatty sequence for sqrt(3): (1,3,5,6,8,10,12,13,15,...). a(n) = (1,5,8,12,...). In the notation in Comments: (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, 35, ...) = A356085. (3) u' ^ v = (3, 6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, 81, ...) = A356086. (4) u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087. 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 *) t1 = Intersection[u, v] (* A346308 *) t2 = Intersection[u, v1] (* A356085 *) t3 = Intersection[u1, v] (* A356086 *) t4 = Intersection[u1, v1] (* A356087 *) PROG (Python) from math import isqrt from itertools import count, islice def A346308_gen(): # generator of terms return filter(lambda n:n == isqrt(3*(isqrt(n**2//3)+1)**2), (isqrt(n*n<<1) for n in count(1))) A346308_list = list(islice(A346308_gen(), 30)) # Chai Wah Wu, Aug 06 2022 CROSSREFS Intersection of A001951 and A022838. Cf. A346308, A347467, A347468, A347469. Cf. A001952, A022838, A054406, A356085, A356086, A356087, A356088 (composites instead of intersections). Sequence in context: A212452 A184915 A108173 * A214858 A186276 A047383 Adjacent sequences: A346305 A346306 A346307 * A346309 A346310 A346311 KEYWORD nonn AUTHOR Clark Kimberling, Sep 11 2021 STATUS approved

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Last modified May 28 07:31 EDT 2023. Contains 362992 sequences. (Running on oeis4.)