A001954 -id:A001954 - cp's OEIS frontend

A191432 Dispersion of ([n*x+1/x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

1, 2, 5, 3, 7, 8, 4, 10, 12, 11, 6, 14, 17, 16, 15, 9, 20, 24, 23, 21, 18, 13, 28, 34, 33, 30, 26, 22, 19, 40, 48, 47, 43, 37, 31, 25, 27, 57, 68, 67, 61, 53, 44, 36, 29, 38, 81, 96, 95, 86, 75, 62, 51, 41, 32, 54, 115, 136, 135, 122, 106, 88, 72, 58, 45, 35, 77, 163, 193, 191, 173, 150, 125, 102, 82, 64, 50, 39

Offset: 1

Clark Kimberling, Jun 03 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.
Examples:
  (1) s=A000040 (the primes), D=A114537, u=A114538.
  (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
  (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.
Conjecture: It appears this sequence is related to the even numbers with odd abundance A088827. Looking at the table format if the columns represent the powers of 2 (starting at 2^1) and the rows represent the squares of odd numbers, then taking the product of a term's row and column gives the n-th term in A088827. Example: A088827(67) = (7^2) * (2^6) = 3136. - John Tyler Rascoe, Jul 12 2022

			Northwest corner:
   1    2    3    4    6    9
   5    7   10   14   20   28
   8   12   17   24   34   48
  11   16   23   33   47   67
  15   21   30   43   61   86

Michel Marcus, Antidiagonals n = 1..100, flattened

Cf. A114537, A035513, A035506, A088827.

Cf. A001953, A001954.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12;  (* r=# rows of T, r1=# rows to show *)
c = 40; c1 = 12;  (* c=# cols of T, c1=# cols to show *)
x = Sqrt[2];
f[n_] := Floor[n*x + 1/x] (* f(n) is complement of column 1 *)
mex[list_] :=  NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
  Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];  TableForm[
Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191432 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191432 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

s(n) = my(x=quadgen(8)); floor(n*x+1/x);  \\ A001953
t(n) = floor((n+1/2)*(2+quadgen(8))); \\ A001954
T(n, k) = my(x = t(n-1)); for (i=2, k, x = s(x);); x; \\ Michel Marcus, Jul 13 2022

A356219 Intersection of A001952 and A003151.

Original entry on oeis.org

284, 287, 289, 292, 294, 296, 299, 301, 304, 306, 309, 311, 313, 316, 318, 321, 323, 325, 328, 330, 333, 335, 337, 340, 342, 345, 347, 350, 352, 354, 357, 359, 362, 364, 366, 369, 371, 374, 376, 379, 381, 383, 386, 388, 391, 393, 395, 398, 400

Offset: 1

Clark Kimberling, Nov 13 2022

This is the third 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.
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 A356219, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).

			(1)  u ^ v = (2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, ...) = A003151.
(2)  u ^ v' = (1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, ...) = A001954.
(3)  u' ^ v = (284, 287, 289, 292, 294, 296, 299, 301, 304, 306, ...) = A356219.
(4)  u' ^ v' = (3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, ...) = A003152.

Cf. A001951, A001952, A003151, A003152, A001954, A184922 (results of compositions instead of intersections), A341239 (reversed compositions).

z = 200;
r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
r1 = 1 + Sqrt[2]; v = Table[Floor[n*r1], {n, 1, z}]  (* A003151 *)
v1 = Take[Complement[Range[1000], v], z]  (* A003152 *)
t1 = Intersection[u, v]    (* A003151 *)
t2 = Intersection[u, v1]   (* A001954 *)
t3 = Intersection[u1, v]   (* A356219 *)
t4 = Intersection[u1, v1]  (* A001952 *)

cp's OEIS Frontend

A191432 Dispersion of ([n*x+1/x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

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