A002389 -id:A002389 - cp's OEIS frontend

A073004 Decimal expansion of exp(gamma).

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

See references and additional links in A094644.
The Riemann hypothesis holds if and only if the inequality sigma(n)/(n*log(log(n))) < exp(gamma) is valid for all n >= 5041, (G. Robin, 1984). - Peter Luschny, Oct 18 2020
From Peter Bala, Aug 24 2025: (Start)
By definition, 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)*exp(s(n+k)). Then it appears that E(n) converges rapidly to exp(gamma). For example, E(50) = 1.78107241799019798523650410310(43...) gives exp(gamma) correct to 29 decimal digits. Cf. A002389. (End)

			Exp(gamma) = 1.7810724179901979852365041031071795491696452143034302053...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.5.1 and 2.27.2, pp. 31, 187.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 166, 191, 208.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
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.
Paul Erdős and S. K. Zaremba, The arithmetic function Sum_{d|n} log d/d, Demonstratio Mathematica, Vol. 6 (1973), pp. 575-579.
T. H. Gronwall, Some Asymptotic Expressions in the Theory of Numbers, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628.
Simon Plouffe, The exp(gamma).
G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Gronwall's Theorem.
Eric Weisstein's World of Mathematics, Mertens Theorem, Equations 2-3.
Eric Weisstein's World of Mathematics, Robin's Theorem.

Cf. A001620 (Euler-Mascheroni constant, gamma).

Cf. A001113, A002389, A067698, A080130, A091901, A094644 (continued fraction for exp(gamma)), A155969, A246499.

R:=RealField(100); Exp(EulerGamma(R)); // G. C. Greubel, Aug 27 2018

RealDigits[ E^(EulerGamma), 10, 110] [[1]]

PARI
```
exp(Euler)
```

By Mertens theorem, equals lim_{m->infinity}(1/log(prime(m))*Product_{k=1..m} 1/(1-1/prime(k))). - Stanislav Sykora, Nov 14 2014
Equals limsup_{n->oo} sigma(n)/(n*log(log(n))) (Gronwall, 1913). - Amiram Eldar, Nov 07 2020
Equals limsup_{n->oo} (Sum_{d|n} log(d)/d)/(log(log(n)))^2 (Erdős and Zaremba, 1973). - Amiram Eldar, Mar 03 2021
Equals Product_{k>=1} (1-1/(k+1))*exp(1/k). - Amiram Eldar, Mar 20 2022
Equals lim_{n->oo} n * Product_{prime p<=n} p^(1/(1-p)). - Thomas Ordowski, Jan 30 2023
Equals Product_{k>=1} (k/sqrt(2))^((-1)^k/(k*log(2))). - Antonio Graciá Llorente, Oct 11 2024
Equals lim_{n->oo} (1/log(n))*Product_{prime p<=n} p/(p - 1) [Mertens] (see Finch at p. 31). - Stefano Spezia, Oct 27 2024

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

Original entry on oeis.org

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.

A059560 Beatty sequence for 1 - 1/log(gamma).

Original entry on oeis.org

2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 31, 33, 36, 39, 42, 45, 47, 50, 53, 56, 59, 62, 64, 67, 70, 73, 76, 78, 81, 84, 87, 90, 93, 95, 98, 101, 104, 107, 109, 112, 115, 118, 121, 124, 126, 129, 132, 135, 138, 140, 143, 146, 149, 152, 155, 157, 160, 163, 166, 169, 172

Offset: 1

Mitch Harris, Jan 22 2001

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

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.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059559.

Cf. A001620, A002389.

seq( floor(n*(1-1/log(gamma))),n=0..100) ;

Floor[Range[100]*(1 - 1/Log[EulerGamma])] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=1 + 1/log(1/Euler); for (n = 1, 2000, write("b059560.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(1-1/log(gamma))). - Michel Marcus, Jan 05 2015

A182500 Decimal expansion of -log_Pi(gamma), where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Volker Werner, May 02 2012

			0.48006024807667318305575522901866575525227806038853...

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

Cf. A001620, A000796.

SetDefaultRealField(RealField(100)); R:= RealField(); -Log(EulerGamma(R))/Log(Pi(R)); // G. C. Greubel, Sep 01 2018

RealDigits[ N[Log[EulerGamma]/Log[Pi], 105]] [[1]]

default(realprecision, 100); -log(Euler)/log(Pi) \\ G. C. Greubel, Sep 01 2018

Equals |A002389/A053510|.

A213440 Decimal expansion of 1 + log(gamma), where gamma is Euler's constant A001620.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 11 2012

			0.4504606870183551776623382311970922116693010187369352...

W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Cf. A001620, A002389.

R:= RealField(100); 1 + Log(EulerGamma(R)); // G. C. Greubel, Aug 27 2018

RealDigits[1 + Log[EulerGamma], 10, 100][[1]] (* G. C. Greubel, Aug 27 2018 *)

default(realprecision, 100); 1 + log(Euler) \\ G. C. Greubel, Aug 27 2018

A059192 Engel expansion of log(1/gamma) (where gamma is the Euler-Mascheroni constant A001620) = 0.549539...

Original entry on oeis.org

2, 11, 12, 13, 53, 348, 5263, 9960, 17040, 33193, 72960, 125350, 663179, 1096815, 3481893, 4802237, 7782503, 9659740, 279957736, 454935116, 460488754, 1710020367, 51367039980, 55286622194, 323648965384, 2061149370731

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, Table of n, a(n) for n = 1..1000
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. A002389.

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[Log[1/EulerGamma], 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)

A182526 Decimal expansion of | log_gamma(e) |, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 1

Volker Werner, May 03 2012

Reciprocal of A002389.

			1.81970602717080047632906605775390654455974544981902...

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

Cf. A001113, A001620, A002389.

SetDefaultRealField(RealField(100)); R:= RealField(); -1/Log(EulerGamma(R)); // G. C. Greubel, Sep 06 2018

RealDigits[1/Log[EulerGamma], 10, 100][[1]]

default(realprecision, 100); -1/log(Euler) \\ G. C. Greubel, Sep 06 2018

Equals 1/A002389.

cp's OEIS Frontend

A073004 Decimal expansion of exp(gamma).

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

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A059560 Beatty sequence for 1 - 1/log(gamma).

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A182500 Decimal expansion of -log_Pi(gamma), where gamma is the Euler-Mascheroni constant.

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A213440 Decimal expansion of 1 + log(gamma), where gamma is Euler's constant A001620.

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A059192 Engel expansion of log(1/gamma) (where gamma is the Euler-Mascheroni constant A001620) = 0.549539...

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Mathematica

A182526 Decimal expansion of | log_gamma(e) |, where gamma is the Euler-Mascheroni constant.

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