A249073 -id:A249073 - cp's OEIS frontend

A249097 Ordered union of the sets {h^6, h >=1} and {3*k^6, k >=1}.

1, 3, 64, 192, 729, 2187, 4096, 12288, 15625, 46656, 46875, 117649, 139968, 262144, 352947, 531441, 786432, 1000000, 1594323, 1771561, 2985984, 3000000, 4826809, 5314683, 7529536, 8957952, 11390625, 14480427, 16777216, 22588608, 24137569, 34012224, 34171875

Offset: 1

Clark Kimberling, Oct 21 2014

Let S = {h^6, h >=1} and T = {3*k^6, k >=1}. Then S and T are disjoint. The position of n^6 in the ordered union of S and T is A249098(n), and the position of 3*n^6 is A249079(n).

			{h^6, h >=1} = {1, 64, 729, 4096, 15625, 46656, 117649, ...};
{3*k^6, k >=1} = {3, 192, 2187, 12288, 46875, 139968, ...};
so the union is {1, 3, 64, 192, 729, 2187, 4096, 12288, ...}

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

Cf. A001014, A249098, A249099, A249073.

upto(n)=setunion(apply(k->k^6, [1..sqrtnint(n,6)]), apply(k->3*k^6, [1..sqrtnint(n\3,6)])) \\ Andrew Howroyd, Feb 18 2025

A249099 Position of 3n^6 in the ordered union of {h^6, h >=1} and {3k^6, k >=1}.

Original entry on oeis.org

2, 4, 6, 8, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 55, 57, 59, 61, 63, 66, 68, 70, 72, 74, 77, 79, 81, 83, 85, 88, 90, 92, 94, 96, 99, 101, 103, 105, 107, 110, 112, 114, 116, 118, 121, 123, 125, 127, 129, 132, 134

Offset: 1

Clark Kimberling, Oct 21 2014

Let S = {h^6, h >=1} and T = {3*k^6, k >=1}. Then S and T are disjoint, with ordered union given by A249097. The position of n^6 is A249098(n), and the position of 3*n^6 is a(n).

Also, a(n) is the position of n in the joint ranking of the positive integers and the numbers k*3^(1/6), so that A249098 and this sequence are a pair of Beatty sequences.

			{h^6, h >=1} = {1, 64, 729, 4096, 15625, 46656, 117649, ...};
{3*k^6, k >=1} = {3, 192, 2187, 12288, 46875, 139968, ...};
so the ordered union is {1, 3, 64, 192, 729, 2187, 4096, 12288, ...}, and
a(2) = 4 because 3*2^6 is in position 4.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A246708, A249073, A249097, A249098.

z = 200; s = Table[h^6, {h, 1, z}]; t = Table[3*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}]]  (* A249098 *)
Flatten[Table[Flatten[Position[v, t[[n]]]], {n, 1, m}]]  (* A249099 *)

a(n) = sqrtnint(3*n^6,6) + n; \\ Kevin Ryde, Feb 18 2025

a(n) = floor((1+3^(1/6)) * n). - Kevin Ryde, Feb 18 2025

Incorrect conjectured formulas removed by Kevin Ryde, Feb 18 2025

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

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

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 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

			{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.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A249073, A249123.

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(2*n^6, 6) \\ David A. Corneth, Aug 11 2019

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

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A249097 Ordered union of the sets {h^6, h >=1} and {3*k^6, k >=1}.

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A249099 Position of 3*n^6 in the ordered union of {h^6, h >=1} and {3*k^6, k >=1}.

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

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

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A249099 Position of 3n^6 in the ordered union of {h^6, h >=1} and {3k^6, k >=1}.

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