A249124 -id:A249124 - cp's OEIS frontend

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

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

Clark Kimberling, Oct 21 2014

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).

			{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, ...}.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A249096, A249123, A249124.

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 *)

A249123 Position of n^6 in the ordered union of {h^6, h >= 1} and {2*k^6, k >= 1}.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 20, 22, 24, 26, 28, 30, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 52, 54, 56, 58, 60, 62, 64, 66, 68, 69, 71, 73, 75, 77, 79, 81, 83, 85, 86, 88, 90, 92, 94, 96, 98, 100, 102, 103, 105, 107, 109, 111, 113, 115, 117, 119

Offset: 1

Clark Kimberling, Oct 21 2014

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 in is A249123(n), and the position of 2*n^6 is A249124(n). Also, a(n) is the position of n 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.

			{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) = 3 because 2^6 is in position 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A249073, A249124.

Res:= NULL: count:= 0:
a:= 1: b:= 1:
for pos from 1 while count < 100 do
  if a^6 < 2*b^6 then
    Res:= Res, pos;
    count:= count+1;
    a:= a+1
  else
    b:= b+1
  fi
od:
Res; # Robert Israel, Aug 11 2019

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 *)

a(n) = n + sqrtnint(((n^6) \ 2), 6) \\ David A. Corneth, Aug 11 2019

a(n) = n + floor(2^(-1/6)*n). - Robert Israel, Aug 12 2019

cp's OEIS Frontend

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

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