cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A249073 Ordered union of the sets {h^6, h >=1} and {2*k^6, k >=1}.

Original entry on oeis.org

1, 2, 64, 128, 729, 1458, 4096, 8192, 15625, 31250, 46656, 93312, 117649, 235298, 262144, 524288, 531441, 1000000, 1062882, 1771561, 2000000, 2985984, 3543122, 4826809, 5971968, 7529536, 9653618, 11390625, 15059072, 16777216, 22781250, 24137569, 33554432
Offset: 1

Views

Author

Clark Kimberling, Oct 21 2014

Keywords

Comments

Let S = {h^6, h >=1} and T = {2*k^6, k >=1}. Then S and T are disjoint. The position of n^6 in the ordered union of S and T is A249123(n), and the position of 2*n^6 is A249124(n).

Examples

			{h^6, h >=1} = {1, 64, 729, 4096, 15625, 46656, 117649, ...};
{2*k^6, k >=1} = {2, 128, 1458, 8192, 31250, 93312, ...};
so the union is {1, 2, 64, 128, 729, 1458, 4096, 8192, 15625, ...}.

Links

Crossrefs

Cf. A249096, A249123, A249124.

Programs

  • Mathematica
    z = 120; s = Table[h^6, {h, 1, z}]; t = Table[2 k^6, {k, 1, z}]; v = Union[s, t]
Flatten[Table[{n^6,2n^6},{n,20}]]//Union (* Harvey P. Dale, Dec 19 2015 *)

A249124 Position of 2*n^6 in the ordered union of {h^6, h >= 1} and {2*k^6, k >= 1}.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 19, 21, 23, 25, 27, 29, 31, 33, 36, 38, 40, 42, 44, 46, 48, 50, 53, 55, 57, 59, 61, 63, 65, 67, 70, 72, 74, 76, 78, 80, 82, 84, 87, 89, 91, 93, 95, 97, 99, 101, 104, 106, 108, 110, 112, 114, 116, 118, 120, 123, 125, 127, 129, 131
Offset: 1

Views

Author

Clark Kimberling, Oct 21 2014

Keywords

Comments

Let S = {h^6, h >= 1} and T = {2*k^6, k >= 1}. Then S and T are disjoint, and their ordered union is given by A249073. The position of n^6 is A249123(n), and the position of 2*n^6 is A249124(n). Also, a(n) is the position of n*2^(1/6) in the joint ranking of the positive integers and the numbers k*2^(1/6), so that A249123 and A249124 are a pair of Beatty sequences.
Every positive integer m is of the form k + floor( (2*k^6)^(1/6) ) (this sequence) or of the form k + floor( (k^6 / 2)^(1/6) ) (A249123) for some positive integer k but not both. - David A. Corneth, Aug 12 2019

Examples

			{h^6, h >= 1} = {1, 64, 729, 4096, 15625, 46656, 117649, ...};
{2*k^6, k >= 1} = {2, 128, 1458, 8192, 31250, 93312, ...};
so the ordered union is {1, 2, 64, 128, 729, 1458, 4096, 8192, 15625, ...}, and
a(2) = 4 because 2*2^6 is in position 4.

Links

Crossrefs

Cf. A249073, A249123.

Programs

  • Mathematica
    z = 200; s = Table[h^6, {h, 1, z}]; t = Table[2*k^6, {k, 1, z}]; u = Union[s, t];
v = Sort[u]  (* A249073 *)
m = Min[120, Position[v, 2*z^2]]
Flatten[Table[Flatten[Position[v, s[[n]]]], {n, 1, m}]]  (* A249123 *)
Flatten[Table[Flatten[Position[v, t[[n]]]], {n, 1, m}]]  (* A249124 *)
  • PARI
    a(n) = n + sqrtnint(2*n^6, 6) \\ David A. Corneth, Aug 11 2019

Formula

a(n) = n + floor( (2*n^6)^(1/6) ). - David A. Corneth, Aug 11 2019
Showing 1-2 of 2 results.

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