A054071 -id:A054071 - cp's OEIS frontend

A054073 Fractal sequence induced by sqrt(2): for k >= 1 let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*sqrt(2)} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),...

1, 1, 2, 3, 1, 2, 3, 1, 4, 2, 5, 3, 1, 4, 2, 5, 3, 1, 6, 4, 2, 5, 3, 1, 6, 4, 2, 7, 5, 3, 8, 1, 6, 4, 2, 7, 5, 3, 8, 1, 6, 4, 9, 2, 7, 5, 10, 3, 8, 1, 6, 4, 9, 2, 7, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 13, 1, 6, 11

Offset: 1

Clark Kimberling

nonn

A054073 generates the interspersion A054077; see A194832 and the Mathematica program.

			p(1)=(1); p(2)=(1,2); p(3)=(3,1,2); p(4)=(3,1,4,2).
When formatted as a triangle, the first 9 rows:
1
1 2
3 1 2
3 1 4 2
5 3 1 4 2
5 3 1 6 4 2
5 3 1 6 4 2 7
5 3 8 1 6 4 2 7
5 3 8 1 6 4 9 2 7

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A054071, A054072, A194832.

r = Sqrt[2];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]],
{n, 1, 20}]] (* A054073 *)
TableForm[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 15}]]
row[n_] := Position[f, n];
u = TableForm[Table[row[n], {n, 1, 20}]]
g[n_, k_] := Part[row[n], k];
p = Flatten[Table[g[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A054077 *)
q[n_] := Position[p, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A054076 *)
(* Clark Kimberling, Sep 03 2011 *)

A077113 Number of nonnegative integer cubes <= n^2.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19

Offset: 0

Reinhard Zumkeller, Oct 29 2002

a(n) is the least number m such that m^3 > n^2. - Zak Seidov, May 03 2005

			Cubes <= 10^2: 0, 1, 8, 27 and 64, hence a(10) = 5.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A026409, A026414, A054071, A077106, A077121, A100196.

Table[Floor[n^(2/3) + 1], {n, 0, 100}] (* Zak Seidov, May 03 2005 *)

from sympy import integer_nthroot
def A077113(n): return integer_nthroot(n**2,3)[0]+1 # Chai Wah Wu, Aug 15 2025

a(n) = floor(n^(2/3))+1.
a(n) = [x^(n^2)] (1/(1 - x))*Sum_{k>=0} x^(k^3). - Ilya Gutkovskiy, Apr 20 2018
a(n) = A100196(n) + 1. - Amiram Eldar, Apr 05 2025

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A054073 Fractal sequence induced by sqrt(2): for k >= 1 let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*sqrt(2)} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica