A007507 -id:A007507 - cp's OEIS frontend

A062979 Continued fraction for 2^sqrt(2), A007507.

2, 1, 1, 1, 72, 3, 4, 1, 3, 2, 1, 1, 1, 14, 1, 2, 1, 1, 3, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 2, 9, 1, 2, 1, 4, 1, 1, 6, 4, 8, 1, 6, 2, 1, 1, 1, 1, 1, 5, 1, 6, 1, 1, 2, 2, 6, 68, 1, 3, 3, 4, 10, 8, 4, 1, 16, 10, 1, 1, 3, 1, 25, 2, 3, 2, 1, 3, 6, 2, 1, 2, 3, 29, 1, 4, 3, 4, 3, 2, 5, 2, 1, 1, 2, 13, 1, 8, 1, 4, 1

Offset: 0

Jason Earls, Jul 24 2001

			2^sqrt(2) = 2.66514414269022... = 2 + 1/(1 + 1/(1 + 1/(1 + 1/(72 + ...)))). - _Harry J. Smith_, Apr 21 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999

Cf. A007507 (decimal expansion).

ContinuedFraction[2^Sqrt[2],100] (* Harvey P. Dale, Oct 16 2012 *)

PARI
```
contfrac(2^(sqrt(2)))
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(2^sqrt(2)); for (n=1, 20000, write("b062979.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 21 2009

Offset changed by Andrew Howroyd, Aug 04 2024

A078333 Decimal expansion of sqrt(2)^sqrt(2).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Nov 21 2002

This number was used in a non-constructive proof that an irrational number raised to an irrational power may be a rational number: "sqrt(2)^sqrt(2) is either rational or irrational. If it is rational, our statement is proved. If it is irrational, (sqrt(2)^sqrt(2))^sqrt(2) = 2 proves our statement." (Jarden, 1953). - Amiram Eldar, Aug 14 2020

			sqrt(2)^sqrt(2) = 1.632526919438152844773495381...

Paul R. Halmos, Problems for mathematicians, young and old, The Mathematical Association of America, 1991. Problem 3 B, pp. 22 and 171.
Dov Jarden, Curiosa No. 339: A simple proof that a power of an irrational number to an irrational exponent may be rational, Scripta Mathematica, Vol. 19 (1953), p. 229.

G. C. Greubel, Table of n, a(n) for n = 1..5000
J. Roger Hindley, The Root-2 Proof as an Example of Non-constructivity, 2015.
J. P. Jones and S. Toporowski, Irrational numbers, American Mathematical Monthly, Vol. 80, No. 4 (1973), pp. 423-424.
Robert Munafo, Notable Properties of Specific Numbers (entry for the number 1.632526919438)
Wikipedia, Square root of the Gelfond-Schneider constant
Index entries for transcendental numbers

Cf. A002193.
Square root of A007507. - Michel Marcus, Oct 21 2017
Cf. A185111 (sqrt(2)^sqrt(3)), A185094 (sqrt(3)^sqrt(3)).

RealDigits[Sqrt[2]^Sqrt[2], 10, 111][[1]]

2^.5^.5 \\ Charles R Greathouse IV, Mar 22 2013

Equals exp(zeta'(1/2, 3) - zeta'(1/2)) = exp((zeta'(-1/2, 3) - zeta'(-1/2))/2), where zeta' is the first derivative of the Hurwitz zeta function and zeta' the first derivative of the Riemann zeta function. - Thomas Scheuerle, Apr 22 2024

Munafo link clarified by Robert Munafo, Jan 25 2010

A038127 A Beatty sequence: a(n) = floor(n*2^sqrt(2)).

Original entry on oeis.org

0, 2, 5, 7, 10, 13, 15, 18, 21, 23, 26, 29, 31, 34, 37, 39, 42, 45, 47, 50, 53, 55, 58, 61, 63, 66, 69, 71, 74, 77, 79, 82, 85, 87, 90, 93, 95, 98, 101, 103, 106, 109, 111, 114, 117, 119, 122, 125, 127, 130, 133, 135, 138, 141, 143, 146, 149, 151

Offset: 0

Felice Russo

nonn

2^sqrt(2) is the Hilbert number (a.k.a. Gelfond-Schneider constant) (A007507).

Of course this is different from A047480.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Hilbert number
Index entries for sequences related to Beatty sequences

[Floor(n*2^(Sqrt(2))): n in [1..50]]; // G. C. Greubel, Mar 27 2018

Floor[2^Sqrt[2] Range[0,60]] (* Harvey P. Dale, Dec 03 2012 *)

for(n=1,50, print1(floor(n*2^(sqrt(2))), ", ")) \\ G. C. Greubel, Mar 27 2018

cp's OEIS Frontend

A062979 Continued fraction for 2^sqrt(2), A007507.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A078333 Decimal expansion of sqrt(2)^sqrt(2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A038127 A Beatty sequence: a(n) = floor(n*2^sqrt(2)).

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI