A246710 -id:A246710 - cp's OEIS frontend

A351208 Decimal expansion of the 11th root of 3.

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

Offset: 1

Mark Andreas, Feb 04 2022

			1.1050315033964...

Index entries for algebraic numbers, degree 11.

Cf. A246710 (8th root), A011446 (9th root), A246711 (10th root).

Maple
```
Digits:=100: evalf(3^(1/11));
```
Mathematica
```
RealDigits[3^(1/11), 10, 108][[1]]
```
PARI
```
sqrtn(3, 11)
```

from sympy import integer_nthroot
def A351208(n): return integer_nthroot(3*10**(11*(n-1)),11)[0] % 10 # Chai Wah Wu, Mar 07 2022

Equals 3^(1/11).

A351209 Decimal expansion of the 12th root of 3.

Original entry on oeis.org

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

Offset: 1

Mark Andreas, Feb 07 2022

			1.0958726911352443801600191280725486527997513...

Index entries for algebraic numbers, degree 12.

Cf. A246710 (8th root), A011446 (9th root), A246711, (10th root), A351208 (11th root).

Maple
```
Digits:=100: evalf(3^(1/12));
```
Mathematica
```
RealDigits[3^(1/12), 10, 108][[1]]
```
PARI
```
sqrtn(3,12)
```

from sympy import integer_nthroot
def A351209(n): return integer_nthroot(3*10**(12*(n-1)),12)[0] % 10 # Chai Wah Wu, Mar 07 2022

Equals 3^(1/12).

A010769 Decimal expansion of 7th root of 2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

This is also the unique positive attractor of the mapping M(x) = sqrt(sqrt(sqrt(2*x))). In general, (p^N-1)-th root of a number f can be approximated by iterating the mapping M(x) = (f*x)^(1/p^N). The convergence is very fast. In this case, p=2, N=3, and f=2. In the form "evaluate the 3rd (or 7th or 15th) root of a number using only square roots", the insight is usable as a recreational math puzzle. - Stanislav Sykora, Oct 26 2015

			1.104089513673812337649505387623...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cube roots (p=2,N=2) for various f: A002580 (2), A002581 (3), A005480 (4), A010582 (10), A092041 (e). 7th roots (p=2,N=3): A246709 (3), A011186 (4), A011201 (5), A011276 (10), A092516 (e). 8th roots (p=3,N=2): A010770 (2), A246710 (3), A011202 (5), A011277 (10). 15th roots (p=2,N=4): A010777(2), A011194(4), A011209(5), A011284(10). - Stanislav Sykora, Oct 26 2015

RealDigits[N[2^(1/7), 100]][[1]] (* Vincenzo Librandi, Apr 02 2013 *)
RealDigits[Surd[2,7],10,120][[1]] (* Harvey P. Dale, Sep 05 2022 *)

sqrtn(2,7) \\ Charles R Greathouse IV, Apr 15 2014

{ default(realprecision, 100); x= 2^(1/7); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Altug Alkan, Nov 14 2015

Equals Product_{k>=0} (1 + (-1)^k/(7*k + 6)). - Amiram Eldar, Jul 29 2020

cp's OEIS Frontend

A351208 Decimal expansion of the 11th root of 3.

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A351209 Decimal expansion of the 12th root of 3.

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A010769 Decimal expansion of 7th root of 2.

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