A098907 -id:A098907 - cp's OEIS frontend

A155969 Decimal expansion of the square of the Euler-Mascheroni constant.

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

Offset: 0

R. J. Mathar, Jan 31 2009

The Pierce expansion is 3, 2144, 2463, 5226, 17239, 51372, 287963, 387316, 3226210,...
From Peter Bala, Aug 24 2025: (Start)
By definition, the Euler-Mascheroni constant gamma = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k - log(n). The convergence is slow. For example, s(50) = 0.5(87...) is only correct to 1 decimal digit. Let S(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). Elsner shows that S(n) converges to gamma much more rapidly. For example, S(50) = 0.57721566490153286060651209008(02...) gives gamma correct to 29 decimal digits.
Define E(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k)^2. Then it appears that E(n) converges rapidly to gamma^2. For example, E(50) = 0.33317792380771867431837613635524(22...) gives gamma^2 correct to 32 decimal digits. (End)

			0.3331779238077186743183761363552442...

G. C. Greubel, Table of n, a(n) for n = 0..1000
C. Elsner, On a sequence transformation with integral coefficients for Euler's constant, Proc. Amer. Math. Soc., Vol. 123 (1995), Number 5, pp. 1537-1541.
Simon Plouffe 100,000 digits

Cf. A001620, A002389, A073004, A098907, A346589

Maple
```
evalf(gamma^2);
```

RealDigits[N[EulerGamma^2, 100]][[1]] (* G. C. Greubel, Dec 26 2016 *)

PARI
```
Euler^2 \\ G. C. Greubel, Dec 26 2016
```

Equals A001620^2.

A147708 Decimal expansion of cosh(EulerGamma).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Nov 11 2008

			1.171265950778541577530323658949030167967677800614291686755912...

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

Cf. A001620, A098907.

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

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

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

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

A147709 Decimal expansion of sinh(EulerGamma).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Nov 11 2008

			Equals 0.6098064672116564077061804441581493812019674136891385186017534...

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

Cf. A001620, A098907.

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

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

First[RealDigits[N[(Exp[EulerGamma] - Exp[ -EulerGamma])/2, 100]]]
RealDigits[Sinh[EulerGamma],10,120][[1]] (* Harvey P. Dale, Mar 06 2013 *)

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

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

Corrected by Harvey P. Dale, Mar 06 2013

A147710 Decimal expansion of tanh(EulerGamma).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Nov 11 2008

			0.52063877277941655882939459166902813428767319381048760826540...

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

Cf. A001620, A098907.

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

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

First[RealDigits[N[(Exp[EulerGamma] - Exp[ -EulerGamma])/(Exp[EulerGamma] + Exp[ -EulerGamma]), 100]]]
RealDigits[Tanh[EulerGamma],10,120][[1]] (* Harvey P. Dale, Aug 10 2020 *)

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

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

A147711 Decimal expansion of coth(EulerGamma).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Nov 11 2008

			1.920717496051102737973648663483211255479106194024976155441...

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

Cf. A001620, A098907.

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

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

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

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

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

A059558 Beatty sequence for 1 + 1/gamma^2.

Original entry on oeis.org

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

Offset: 1

Mitch Harris, Jan 22 2001

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

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

Beatty complement is A059557.

Cf. A001620, A098907, A155969.

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

{ 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

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

Removed incorrect comment, Joerg Arndt, Nov 14 2014

A059191 Engel expansion of 1/gamma, (gamma is the Euler-Mascheroni constant A001620) = 1.73245.

Original entry on oeis.org

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

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

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

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

Cf. A098907.

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 *)

A172527 a(n) = the smallest prime > (1/EulerGamma)^n.

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Nov 21 2010

nonn

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

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

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

Cf. A001620 A098907.

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

A173647 Primes found in decimal expansion of 1/EulerGamma.

Original entry on oeis.org

17, 173, 173245471460063, 1732454714600633

Offset: 1

Michel Lagneau, Nov 24 2010

Primes found in A098907.

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

Cf. A001620 A098907

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:

A342934 Decimal expansion of gamma^(1/gamma), where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Christoph B. Kassir, Aug 30 2021

			0.385948254719841058037365008117537208453571562501405965467694054181966575...

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

Cf. A001620, A098907, A345066, A346589, A073004, A182547, A182470, A073018.

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

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

PARI
```
Euler^(1/Euler)
```

cp's OEIS Frontend

A155969 Decimal expansion of the square of the Euler-Mascheroni constant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A147708 Decimal expansion of cosh(EulerGamma).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A147709 Decimal expansion of sinh(EulerGamma).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A147710 Decimal expansion of tanh(EulerGamma).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A147711 Decimal expansion of coth(EulerGamma).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

A059558 Beatty sequence for 1 + 1/gamma^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A059191 Engel expansion of 1/gamma, (gamma is the Euler-Mascheroni constant A001620) = 1.73245.

Views

Author

Keywords

Comments

References

Links