A005534 -id:A005534 - cp's OEIS frontend

A002951 Continued fraction for fifth root of 5.

1, 2, 1, 1, 1, 2, 1, 2, 8, 1, 25, 1, 5, 1, 22, 1, 8, 1, 1, 9, 1, 1, 4, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 6, 2, 46, 1, 12, 1, 32, 1, 2, 3, 2, 3, 55, 1, 12, 3, 8, 1, 1, 11, 1, 4, 1, 1, 1, 2, 1, 1, 7, 1, 1, 4, 3, 3, 3218, 1, 3, 1, 2, 2, 3, 1, 1, 2, 11, 1, 7, 57, 2, 2, 2, 2, 1, 1, 67, 1, 2, 3, 1, 1, 13, 3

Offset: 0

N. J. A. Sloane

Fifth root of 5 = 5^(1/5). - Harry J. Smith, May 10 2009

			1.379729661461214832390063464... = 1 + 1/(2 + 1/(1 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, May 10 2009

H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
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 = 0..19999
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
J. Shallit & N. J. A. Sloane, Correspondence 1974-1975
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005534 (decimal expansion).

Cf. A002363, A002364 (convergents).

SetDefaultRealField(RealField(100)); ContinuedFraction(5^(1/5)); // G. C. Greubel, Nov 02 2018

with(numtheory): cfrac(5^(1/5),100,'quotients'); # Muniru A Asiru, Nov 02 2018

ContinuedFraction[5^(1/5), 100] (* G. C. Greubel, Nov 02 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(5^(1/5)); for (n=1, 20000, write("b002951.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 10 2009

More terms copied from Smith's b-file by Hagen von Eitzen, Jul 20 2009

Offset changed by Andrew Howroyd, Jul 05 2024

A002363 Denominators of continued fraction convergents to fifth root of 5.

Original entry on oeis.org

1, 2, 3, 5, 8, 21, 29, 79, 661, 740, 19161, 19901, 118666, 138567, 3167140, 3305707, 29612796, 32918503, 62531299, 595700194, 658231493, 1253931687, 5673958241, 6927889928, 19529738097, 26457628025, 72444994147, 98902622172, 270250238491, 639403099154

Offset: 0

N. J. A. Sloane

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..1996
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

Numerators in A002364.

Cf. A002951, A005534.

numtheory:-cfrac(5^(1/5),1000,'con'):
map(denom,con); # Robert Israel, May 13 2018

Denominator[Convergents[Surd[5,5],30]] (* Harvey P. Dale, Jul 16 2016 *)

More terms from Herman P. Robinson
More terms from Harvey P. Dale, Jul 16 2016
Offset changed by Andrew Howroyd, Jul 05 2024

A002364 Numerators of continued fraction convergents to fifth root of 5.

Original entry on oeis.org

1, 3, 4, 7, 11, 29, 40, 109, 912, 1021, 26437, 27458, 163727, 191185, 4369797, 4560982, 40857653, 45418635, 86276288, 821905227, 908181515, 1730086742, 7828528483, 9558615225, 26945758933, 36504374158, 99954507249, 136458881407, 372872270063, 882203421533

Offset: 0

N. J. A. Sloane, Herman P. Robinson

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

Cf. A002363 (denominators), A002951 A005534.

Numerator[Convergents[Surd[5,5],30]] (* Harvey P. Dale, Dec 06 2015 *)

Definition clarified by and more terms from Harvey P. Dale, Dec 06 2015

Offset changed by Andrew Howroyd, Jul 05 2024

A011100 Decimal expansion of 5th root of 15.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			1.7187719275874787770135214520444...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

15^(1/5); // Vincenzo Librandi, Feb 16 2015

RealDigits[15^(1/5), 10, 105][[1]] (* Alonso del Arte, Feb 15 2015 *)

sqrtn(15,5) \\ Charles R Greathouse IV, Apr 16 2014

Equals A005532 * A005534 . - R. J. Mathar, May 22 2024

A011130 Decimal expansion of 5th root of 45.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 1..1000

sqrtn(45,5) \\ Charles R Greathouse IV, Apr 16 2014

Equals A011094*A005534. - R. J. Mathar, Dec 17 2024

A337840 a(n) is the decimal place of the start of the first occurrence of n in the decimal expansion of n^(1/n).

Original entry on oeis.org

0, 4, 10, 1, 38, 6, 9, 4, 12, 17, 26, 0, 264, 144, 107, 101, 101, 4, 78, 68, 36, 86, 11, 17, 147, 151, 205, 50, 55, 26, 307, 88, 94, 180, 177, 61, 113, 244, 280, 37, 110, 38, 285, 101, 124, 223, 243, 25, 86, 116, 66, 77, 146, 283, 3, 60, 20, 82, 27, 146, 82, 140

Offset: 1

William Phoenix Marcum, Sep 25 2020

Does a(n) exist for all n? Some relatively large values: a(1021) = 67714, a(1111) = 64946. - Chai Wah Wu, Oct 07 2020

			For n = 1, 1^(1/1) = 1.0000000, so a(1) is 0.
For n = 12, 12^(1/12) ~= 1.2300755, so a(12) = 0.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A177715.

Decimal expansions of some n^(1/n): A002193, A002581, A005534, A011215, A011231, A011247, A011263, A011279, A011295, A011311, A011327, A011343, A011359.

max = 3000; a[n_] := SequencePosition[RealDigits[n^(1/n), 10, max][[1]], IntegerDigits[n]][[1, 1]] - 1; Array[a, 100] (* Amiram Eldar, Sep 25 2020 *)

a(n) = {if (n==1, 0, my(p=10000); default(realprecision, p+1); my(x = floor(10^p*n^(1/n)), d = digits(x), nb = #Str(n)); for(k=1, #d-nb+1, my(v=vector(nb, i, d[k+i-1])); if (fromdigits(v) == n, return(k-1));); error("not found"););} \\ Michel Marcus, Sep 30 2020

import gmpy2
from gmpy2 import mpfr, digits, root
gmpy2.get_context().precision=10**5
def A337840(n): # increase precision if -1 is returned
    return digits(root(mpfr(n),n))[0].find(str(n)) # Chai Wah Wu, Oct 07 2020

More terms from Amiram Eldar, Sep 25 2020

A281143 Decimal expansion of 10!^(1/10).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jan 15 2017

Base b such that log_b 10! = 10.

Inspired by the idea of utilizing the log scaled to 10! being 10, i.e., log_b 10! = 10, therefore b = 2^(4/5)*3^(2/5)*5^(1/5)*7^(1/10).

			4.52872868811676476220330933719550879349863167608939046288611476046926255...

Cf. A002193, A005486, A005534, A011020, A011092, A011094, A011101.

RealDigits[2^(4/5) 3^(2/5) 5^(1/5) 7^(1/10), 10, 111][[1]] (* or *)
RealDigits[Solve[Log[b, 10!] == 10, b][[1, 1, 2]], 10, 105][[1]]

Equals A011101 * A011094 * A005534 * A011092.

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A002951 Continued fraction for fifth root of 5.

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A002363 Denominators of continued fraction convergents to fifth root of 5.

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A002364 Numerators of continued fraction convergents to fifth root of 5.

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A011100 Decimal expansion of 5th root of 15.

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A011130 Decimal expansion of 5th root of 45.

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A337840 a(n) is the decimal place of the start of the first occurrence of n in the decimal expansion of n^(1/n).

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A281143 Decimal expansion of 10!^(1/10).

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