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A356088
a(n) = A001951(A022838(n)).
12
1, 4, 7, 8, 11, 14, 16, 18, 21, 24, 26, 28, 31, 33, 35, 38, 41, 43, 45, 48, 50, 53, 55, 57, 60, 63, 65, 67, 70, 72, 74, 77, 80, 82, 84, 87, 90, 91, 94, 97, 100, 101, 104, 107, 108, 111, 114, 117, 118, 121, 124, 127, 128, 131, 134, 135, 138, 141, 144, 145
OFFSET
1,2
COMMENTS
This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) u' o v'.
Every positive integer is in exactly one of the four sequences.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo) w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356088, 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.
EXAMPLE
(1) u o v = (1, 4, 7, 8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088.
(2) u o v' = (2, 5, 9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089.
(3) u' o v = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090.
(4) u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091.
MATHEMATICA
z = 600; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}]; (* A001951 *)
u1 = Complement[Range[Max[u]], u]; (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}]; (* A022838 *)
v1 = Complement[Range[Max[v]], v]; (* A054406 *)
Table[u[[v[[n]]]], {n, 1, zz}]; (* A356088 *)
Table[u[[v1[[n]]]], {n, 1, zz}]; (* A356089) *)
Table[u1[[v[[n]]]], {n, 1, zz}]; (* A356090 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]; (* A356091 *)
PROG
(Python)
from math import isqrt
def A356088(n): return isqrt(isqrt(3*n*n)**2<<1) # Chai Wah Wu, Aug 06 2022
CROSSREFS
Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356089, A356090, A356091.
Sequence in context: A288479 A047347 A188376 * A137362 A190765 A024621
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 04 2022
STATUS
approved