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A346308
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Intersection of Beatty sequences for sqrt(2) and sqrt(3).
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17
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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
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OFFSET
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1,2
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COMMENTS
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Let d(n) = a(n) - A022840(n). Conjecture: (d(n)) is unbounded below and above, and d(n) = 0 for infinitely many n.
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)
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LINKS
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FORMULA
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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)).
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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