A246708 -id:A246708 - cp's OEIS frontend

A005532 Decimal expansion of fifth root of 3.

1, 2, 4, 5, 7, 3, 0, 9, 3, 9, 6, 1, 5, 5, 1, 7, 3, 2, 5, 9, 6, 6, 6, 8, 0, 3, 3, 6, 6, 4, 0, 3, 0, 5, 0, 8, 0, 9, 3, 9, 3, 0, 9, 9, 9, 3, 0, 6, 8, 7, 7, 9, 8, 1, 1, 0, 4, 6, 1, 7, 3, 0, 1, 4, 3, 6, 0, 7, 4, 6, 6, 5, 3, 7, 7, 5, 4, 9, 3, 5, 6, 6, 6, 0, 5, 8, 9, 5, 1, 4, 4, 5, 8, 8, 1, 2, 3, 4, 2, 5, 6, 5, 9, 0, 2

Offset: 1

N. J. A. Sloane

			1.245730939615517325966680336640305080939309993068779811046173... - _Harry J. Smith_, May 12 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 5.

Cf. A003117 (continued fraction). - Harry J. Smith, May 12 2009

Cf. A246708.

Digits := 200: it := evalf(3^(1/5)/10, 200)-floor(evalf(3^(1/5)/10, 200)): for i from 1 to 150 do printf(`%d,`,floor(10*it)): it := 10*it-floor(10*it): od:

RealDigits[N[3^(1/5),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2012 *)
RealDigits[Surd[3,5],10,120][[1]] (* Harvey P. Dale, Dec 10 2024 *)

{ default(realprecision, 20080); x=3^(1/5); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b005532.txt", n, " ", d)); } \\ Harry J. Smith, May 12 2009

More terms from James Sellers, Feb 19 2001

A246709 Decimal expansion of the seventh root of 3.

Original entry on oeis.org

1, 1, 6, 9, 9, 3, 0, 8, 1, 2, 7, 5, 8, 6, 8, 6, 8, 8, 6, 4, 6, 2, 9, 7, 5, 7, 2, 5, 5, 1, 3, 7, 3, 4, 6, 6, 7, 6, 9, 9, 4, 0, 4, 1, 9, 6, 4, 2, 0, 9, 3, 4, 2, 0, 9, 0, 3, 0, 2, 1, 8, 9, 6, 5, 5, 8, 9, 3, 3, 3, 9, 3, 6, 2, 7, 4, 0, 0, 3, 6, 7, 0, 1, 5, 1, 1, 4, 3, 0, 1, 9, 6, 9, 3, 9, 3, 5

Offset: 1

Alonso del Arte, Sep 01 2014

			3^(1/7) = 1.1699308127586868864629757255137346676994...

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Szabolcs Tengely and Maciej Ulas, On a problem of Petho, arXiv:1702.06068 [math.NT], 2017.
Index entries for sequences related to the decimal expansion of roots.

Cf. A246708 (sixth root of 3), A011267 (13th root of 9).

Cf. A010769.

Mathematica
```
RealDigits[3^(1/7), 10, 108][[1]]
```

sqrtn(3, 7) \\ Stefano Spezia, Apr 03 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

A382197 Decimal expansion of 24^(1/6).

Original entry on oeis.org

1, 6, 9, 8, 3, 8, 1, 3, 2, 9, 5, 6, 4, 9, 5, 2, 7, 8, 4, 9, 1, 2, 5, 6, 4, 5, 2, 4, 6, 5, 9, 7, 4, 9, 3, 6, 0, 2, 0, 3, 5, 0, 0, 0, 9, 0, 3, 3, 5, 9, 7, 1, 4, 4, 8, 9, 0, 4, 1, 0, 6, 1, 6, 1, 9, 6, 9, 5, 4, 9, 3, 2, 0, 1, 3, 8, 0, 8, 9, 0, 0, 9, 2, 7, 8, 1, 3, 6, 7, 0, 0, 3, 4, 1, 9, 8, 8, 0, 2, 1

Offset: 1

Stefano Spezia, Mar 18 2025

			1.69838132956495278491256452465974936020350009...

Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See Theorem 10 at p. 20.

Cf. A002193, A010480, A010596, A011020, A011109, A246708.

Mathematica
```
RealDigits[24^(1/6),10,100][[1]]
```

Equals A002193*A246708.

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A005532 Decimal expansion of fifth root of 3.

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A246709 Decimal expansion of the seventh root of 3.

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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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A382197 Decimal expansion of 24^(1/6).

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