A098907 -id:A098907

A147708

Decimal expansion of cosh(EulerGamma).

+10
5

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

EXAMPLE

1.171265950778541577530323658949030167967677800614291686755912...

MATHEMATICA

First[RealDigits[N[(Exp[EulerGamma] + Exp[ -EulerGamma])/2, 100]]]

PROG

(PARI) default(realprecision, 100); cosh(Euler) \\ G. C. Greubel, Aug 29 2018

(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Cosh(EulerGamma(R)); // G. C. Greubel, Aug 29 2018

CROSSREFS

Cf. A001620, A098907.

Cf. A147709 (with sinh), A147710 (with tanh), A147711 (with coth).

KEYWORD

cons,nonn

AUTHOR

Artur Jasinski, Nov 11 2008

STATUS

approved

A147709

Decimal expansion of sinh(EulerGamma).

+10
5

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

EXAMPLE

Equals 0.6098064672116564077061804441581493812019674136891385186017534...

MATHEMATICA

First[RealDigits[N[(Exp[EulerGamma] - Exp[ -EulerGamma])/2, 100]]]

RealDigits[Sinh[EulerGamma], 10, 120][[1]] (* Harvey P. Dale, Mar 06 2013 *)

PROG

(PARI) default(realprecision, 100); (exp(Euler) - exp(-Euler))/2 \\ G. C. Greubel, Aug 29 2018

(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (Exp(EulerGamma(R)) - Exp(-EulerGamma(R)))/2; // G. C. Greubel, Aug 29 2018

CROSSREFS

Cf. A001620, A098907.

Cf. A147708 (with cosh), A147710 (with tanh), A147711 (with coth).

KEYWORD

cons,nonn

AUTHOR

Artur Jasinski, Nov 11 2008

EXTENSIONS

Leading zero removed, offset adjusted by R. J. Mathar, Feb 05 2009

Corrected by Harvey P. Dale, Mar 06 2013

STATUS

approved

A147710

Decimal expansion of tanh(EulerGamma).

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

EXAMPLE

0.52063877277941655882939459166902813428767319381048760826540...

MATHEMATICA

First[RealDigits[N[(Exp[EulerGamma] - Exp[ -EulerGamma])/(Exp[EulerGamma] + Exp[ -EulerGamma]), 100]]]

RealDigits[Tanh[EulerGamma], 10, 120][[1]] (* Harvey P. Dale, Aug 10 2020 *)

PROG

(PARI) default(realprecision, 100); tanh(Euler) \\ G. C. Greubel, Aug 29 2018

(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Tanh(EulerGamma(R)); // G. C. Greubel, Aug 29 2018

CROSSREFS

Cf. A001620, A098907.

Cf. A147708 (with cosh), A147709 (with sinh), A147711 (with coth).

KEYWORD

cons,nonn

AUTHOR

Artur Jasinski, Nov 11 2008

EXTENSIONS

Leading zero removed and offset adjusted by R. J. Mathar, Feb 05 2009

STATUS

approved

A147711

Decimal expansion of coth(EulerGamma).

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

EXAMPLE

1.920717496051102737973648663483211255479106194024976155441...

MATHEMATICA

First[RealDigits[N[(Exp[EulerGamma] + Exp[ -EulerGamma])/(Exp[EulerGamma] - Exp[ -EulerGamma]), 100]]]

PROG

(PARI) default(realprecision, 100); (exp(Euler) + exp(-Euler))/(exp(Euler) - exp(-Euler)) \\ G. C. Greubel, Aug 29 2018

(PARI) 1/tanh(Euler) \\ Charles R Greathouse IV, May 14 2019

(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (Exp(EulerGamma(R)) + Exp(-EulerGamma(R)))/(Exp(EulerGamma(R)) - Exp(-EulerGamma(R))); // G. C. Greubel, Aug 29 2018

CROSSREFS

Cf. A001620, A098907.

Cf. A147708 (with cosh), A147709 (with sinh), A147710 (with tanh).

KEYWORD

cons,nonn

AUTHOR

Artur Jasinski, Nov 11 2008

STATUS

approved

A059558

Beatty sequence for 1 + 1/gamma^2.

+10
3

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The first term where this sequence breaks the progression a(n) = a(n-1) + 4 is a(715) = 2861. - Max Alekseyev, Mar 03 2007

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..2000

Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.

Tanya Khovanova, Non Recursions

Eric Weisstein's World of Mathematics, Beatty Sequence.

Index entries for sequences related to Beatty sequences

FORMULA

a(n) = floor(n*(1+1/gamma^2)) where 1+1/gamma^2= 1+A098907^2 = 4.00139933... - R. J. Mathar, Sep 29 2023

MATHEMATICA

Floor[Range[100]*(1 + 1/EulerGamma^2)] (* Paolo Xausa, Jul 05 2024 *)

PROG

(PARI) { default(realprecision, 100); b=1 + 1/Euler^2; for (n = 1, 2000, write("b059558.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

CROSSREFS

Beatty complement is A059557.

Cf. A001620, A098907, A155969.

KEYWORD

nonn,easy

AUTHOR

Mitch Harris, Jan 22 2001

EXTENSIONS

Removed incorrect comment, Joerg Arndt, Nov 14 2014

STATUS

approved

A059191

Engel expansion of 1/gamma, (gamma is the Euler-Mascheroni constant A001620) = 1.73245.

+10
1

1, 2, 3, 3, 6, 10, 20, 46, 226, 1836, 3719, 14308, 17262, 129530, 945152, 1535786, 2229882, 3560447, 9434930, 20957352, 102311436, 312567415, 449243761, 4362956254, 12000988888, 22909186976, 29969826721

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Cf. A006784 for definition of Engel expansion.

REFERENCES

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

LINKS

G. C. Greubel and T. D. Noe, Table of n, a(n) for n = 1..1000[Terms 1 to 300 computed by T. D. Noe; Terms 301 to 1000 computed by G. C. Greubel, Dec 27 2016]

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.

P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.

Index entries for sequences related to Engel expansions

MATHEMATICA

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@

NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];

EngelExp[N[EulerGamma^2, 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)

CROSSREFS

Cf. A098907.

KEYWORD

nonn,easy,nice

AUTHOR

Mitch Harris

STATUS

approved

A172527

a(n) = the smallest prime > (1/EulerGamma)^n.

+10
0

2, 5, 7, 11, 17, 29, 47, 83, 149, 251, 431, 733, 1277, 2203, 3803, 6599, 11411, 19777, 34253, 59333, 102793, 178067, 308489, 534431, 925891, 1604021, 2778901, 4814321, 8340593, 14449651, 25033357, 43369111, 75135077, 130168021, 225510203

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

EulerGamma is Euler's constant (or the Euler-Mascheroni constant) gamma (A001620).

1/EulerGamma = 1.7324547146006... (A098907).

LINKS

Table of n, a(n) for n=1..35.

EXAMPLE

The first prime > (1/EulerGamma)^6 = 27.03779975... is 29, so a(6) = 29.

MATHEMATICA

Table[Prime[PrimePi[1/EulerGamma^n] + 1], {n, 1, 40}]

NextPrime/@Table[1/EulerGamma^n, {n, 40}] (* Harvey P. Dale, May 10 2020 *)

CROSSREFS

Cf. A001620 A098907.

KEYWORD

nonn

AUTHOR

Michel Lagneau, Nov 21 2010

STATUS

approved

A173647

Primes found in decimal expansion of 1/EulerGamma.

+10
0

17, 173, 173245471460063, 1732454714600633

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Primes found in A098907.

LINKS

Table of n, a(n) for n=1..4.

EXAMPLE

1/EulerGamma =1.732454714600633473583... so a(1)=17 ; a(2) =173,...

MAPLE

Digits := 100; n0 := evalf(1/gamma); for i from 1 to 500 do x := trunc(10^i*n0):

if isprime(x) then printf(`%d, `, x): fi: od:

CROSSREFS

Cf. A001620 A098907

KEYWORD

nonn,base

AUTHOR

Michel Lagneau, Nov 24 2010

STATUS

approved

A342934

Decimal expansion of gamma^(1/gamma), where gamma is the Euler-Mascheroni constant.

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..89.

Michael Ian Shamos, Shamos's catalog of the real numbers, (2011).

EXAMPLE

0.385948254719841058037365008117537208453571562501405965467694054181966575...

MAPLE

Digits := 100; evalf(gamma^(1/gamma));

MATHEMATICA

RealDigits[EulerGamme^(1/EulerGamma), 10, 100][[1]]