A011003 -id:A011003 - cp's OEIS frontend

A234522 Decimal expansion of 7^(1/4) - 5^(1/4).

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

Offset: 0

Jaroslav Krizek, Jan 14 2014

Decimal expansion of maximal value of function beta(n) = sigma(n)^(1/n) - (n+1)^(1/n) for n = 4, where beta(n) is called the beta-deviation from primality of number n (see A234520). Lim_n->infinity beta(n) = 0.

An algebraic integer with degree 16 and minimal polynomial x^16 - 48x^12 - 3896x^8 - 53952x^4 + 16. - Charles R Greathouse IV, Apr 25 2016

			0.13122778047656520129933335...

Index entries for algebraic numbers, degree 16

Cf. A234515, A234516, A234517, A234518, A234519, A234520, A234521, A234523, A234524.

RealDigits[N[7^(1/4)-5^(1/4), 100]][[1]] (* Georg Fischer, Apr 04 2020 *)
RealDigits[Surd[7,4]-Surd[5,4],10,120][[1]] (* Harvey P. Dale, Mar 23 2025 *)

7^(1/4) - 5^(1/4) \\ Charles R Greathouse IV, Apr 25 2016

Equals A011005-A011003.

a(97) corrected by Georg Fischer, Apr 04 2020

A011072 Decimal expansion of 4th root of 80.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.99069756244244108382379798828182679072691951522941...

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

Cf. A011003.

RealDigits[Surd[80,4],10,120][[1]] (* Harvey P. Dale, Nov 02 2015 *)

sqrtn(80,4) \\ Charles R Greathouse IV, Apr 16 2014

Equals 2 * A011003. - Amiram Eldar, May 25 2023

A159824 Continued fraction for Pi^Pi (cf. A073233).

Original entry on oeis.org

36, 2, 6, 9, 2, 1, 2, 5, 1, 1, 6, 2, 1, 291, 1, 38, 50, 1, 2, 5, 4, 1, 2, 2, 1, 5, 1, 4, 13, 2, 1, 4, 3, 3, 1, 2, 25, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 1, 43, 1, 2, 7, 3, 1, 1, 1, 2, 4, 2, 1, 1, 3, 1, 3, 3, 2, 2, 16, 3, 5, 2, 1, 5, 2, 1, 10, 1, 1, 3, 1, 13, 1, 1, 3, 1, 10, 4, 1, 1, 1, 38, 1, 2, 2, 1, 1, 3

Offset: 0

Harry J. Smith, Apr 30 2009

nonn

			36.4621596072079117709908260... = 36 + 1/(2 + 1/(6 + 1/(9 + 1/(2 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A001076, A001203, A001204, A002945, A010767, A011002, A011003, A073233, A073238, A179613, A179615, A179616, A179617.

ContinuedFraction[Pi^Pi,200] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2010 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^Pi); for (n=1, 20001, write("b159824.txt", n-1, " ", x[n])); }

Edited by N. J. A. Sloane, Jul 22 2010

A179616 Continued fraction for 5^(1/4).

Original entry on oeis.org

1, 2, 53, 4, 96, 2, 1, 6, 2, 2, 2, 6, 1, 4, 1, 49, 17, 2, 3, 2, 7, 2, 1, 1, 3, 1, 3, 2, 1, 2, 6, 11, 1, 8, 1, 1, 10, 1, 10, 2, 5, 1, 2, 1, 1, 4, 1, 3, 3, 2, 3, 5, 2, 1, 1, 1, 1, 1, 2, 4, 1, 3, 1, 1, 2, 5, 1, 4, 3, 1, 97, 2, 1, 176, 2, 1, 1, 11, 2, 2, 9, 5, 1, 5, 5, 3, 4, 1, 2, 1, 10, 1, 3, 1, 7, 2, 2, 1, 32

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 20 2010

5^(1/4) = 1.4953487812212205419118989941409133953634...

Cf. A011003 (decimal expansion), A179613, A179615.

Mathematica
```
ContinuedFraction[5^(1/4),200]
```

Offset changed by Andrew Howroyd, Jul 07 2024

A269430 Decimal expansion of (1 + Pi)/2.

Original entry on oeis.org

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

Offset: 1

Jani Melik, Feb 26 2016

			2.0707963267948966192313216916397514420985846996875529...

Index entries for transcendental numbers

Cf. A000796, A002193, A002194, A002163, A174968, A001622, A002161, A010767, A011002, A011003, A191502, A053004.

PARI
```
(1 + Pi)/2 \\ Altug Alkan, Apr 07 2016
```
Sage
```
N((1+pi)/2, digits=110)
```

Equals A096444 + 1 or A019669 + 1/2.

Equals Sum_{k>=0} 2^k/binomial(2*k+2,k). - Amiram Eldar, Jun 30 2020

A344362 Decimal expansion of (5^(1/4) + 5^(-1/4))/2.

Original entry on oeis.org

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

Offset: 1

Daniel Carter, May 15 2021

Solution for x in the system {x = 1/y + 1/z, y = x + 1/z, z = y + 1/x}. The corresponding values of y and z are 5^(1/4) and (5^(1/4) + 5^(3/4))/2.

The smallest aspect ratio of a set of three rectangles which have the property that any two of them can be scaled, rotated, and joined at an edge to obtain a rectangle with the third aspect ratio. The other two aspect ratios are given in the comment above.

			1.082044543098821282957566033699780665875...

Cf. A011003, A344363, A293409, A176015.

RealDigits[Cosh[Log[5]/4], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

solve(x=1, 2, 5*x^4 - 5*x^2 - 1) \\ Hugo Pfoertner, May 16 2021

my(c=50+30*quadgen(20)); a_vector(len) = digits(sqrtint(floor(c*100^(len-2)))); \\ Kevin Ryde, May 28 2021

Equals 5*A293409.
Equals sqrt(A176015).
Equals cosh(log(5)/4). - Vaclav Kotesovec, May 28 2021

A344363 Decimal expansion of (5^(1/4) + 5^(3/4))/2.

Original entry on oeis.org

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

Offset: 1

Daniel Carter, May 15 2021

Solution for z in the system {x = 1/y + 1/z, y = x + 1/z, z = y + 1/x}. The corresponding values of x and y are (5^(1/4) + 5^(-1/4))/2 and 5^(1/4).

The largest aspect ratio of a set of three rectangles which have the property that any two of them can be scaled, rotated, and joined at an edge to obtain a rectangle with the third aspect ratio. The other two aspect ratios are given in the comment above.

			2.419525153051665330964032180217076538651...

I. J. Zucker, G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 eq (8.8)

Cf. A011003, A344362.

RealDigits[(Surd[5,4]+Surd[5^3,4])/2,10,120][[1]] (* Harvey P. Dale, Jan 01 2023 *)

my(c=250+150*quadgen(20)); a_vector(len) = digits(sqrtint(floor(c*100^(len-2)))); \\ Kevin Ryde, May 28 2021

Equals sqrt(A090550).

Equals Gamma(1/20)*Gamma(9/20)/(Gamma(3/20)*Gamma(7/20)). [Zucker] - R. J. Mathar, Jun 24 2024

cp's OEIS Frontend

A234522 Decimal expansion of 7^(1/4) - 5^(1/4).

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A011072 Decimal expansion of 4th root of 80.

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A159824 Continued fraction for Pi^Pi (cf. A073233).

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A179616 Continued fraction for 5^(1/4).

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A269430 Decimal expansion of (1 + Pi)/2.

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A344362 Decimal expansion of (5^(1/4) + 5^(-1/4))/2.

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A344363 Decimal expansion of (5^(1/4) + 5^(3/4))/2.

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