A155969 -id:A155969 - 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

A002389 Decimal expansion of -log(gamma), where gamma is Euler's constant A001620.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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)*log(s(n+k)). Then it appears that E(n) converges rapidly to log(gamma). For example, E(50) = -0.549539312981644822337661768802(88...) gives log(gamma) correct to 30 decimal digits. Cf. A073004. (End)

			.549539312981644822337661768802907788330698981263...

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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ivan Panchenko, 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, -log(gamma) to 10000 digits
Simon Plouffe, -log(gamma) to 1024 digits

Cf. A001620, A073004, A155969, A213440.

SetDefaultRealField(RealField(100)); R:= RealField(); -Log(EulerGamma(R)); // G. C. Greubel, Sep 07 2018

RealDigits[-Log[EulerGamma], 10, 100][[1]] (* G. C. Greubel, Sep 07 2018 *)

-log(Euler) \\ Michel Marcus, Mar 11 2013

A081855 Decimal expansion of Gamma''(1).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Apr 11 2003

Also the decimal expansion of the Integral_{x>=0} exp(-x)*(log(x))^2 dx. - Robert G. Wilson v, Aug 18 2017

			1.978111990655945110790791303001269415878367... [corrected by _Georg Fischer_, Jul 29 2021]

Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 179.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.2, p. 31.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), p. 223-232.

Cf. A001620, A013661, A155969, A261509.

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); EulerGamma(R)^2 + Evaluate(L,2); // G. C. Greubel, Aug 29 2018

EulerGamma^2 + Zeta[2] // RealDigits[#, 10, 105] & // First (* Jean-François Alcover, Apr 29 2013 *)
RealDigits[ Integrate[ Exp[-x]*Log[x]^2, {x, 0, Infinity}], 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2017 *)

Euler^2+zeta(2) \\ Charles R Greathouse IV, Aug 18 2017

intnum(x=0,[oo,1],exp(-x)*log(x)^2) \\ Charles R Greathouse IV, Aug 18 2017

The second derivative of Gamma(x) at x=1 is Gamma^2+zeta(2) = 1.97811199... where Gamma is the Euler constant and zeta(2) = Pi^2/6.

A344964 Decimal expansion of the sum of the reciprocals of the squares of the zeros of the digamma function.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 03 2021

The sum is Sum_{k>=0} 1/x_k^2, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.

			5.26798012435239798373562163629331979431626684387002...

István Mező and Michael E. Hoffman, Zeros of the digamma function and its Barnes G-function analogue, Integral Transforms and Special Functions, Vol. 28, No. 11 (2017), pp. 846-858.
Wikipedia, Digamma function.

Cf. A344965, A344966, A344967, A344968.
Cf. A001620, A102753.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^2/2 + EulerGamma^2, 10, 100][[1]]

Equals Pi^2/2 + gamma^2 = A102753 + A155969, where gamma is Euler's constant (A001620).

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

A070860 Decimal expansion of Pi^2/12 - gamma^2 /2.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, May 24 2003

This is erroneously computed as the linear term in the Laurent expansion of Gamma(x) = 1/x +c(0) + c(1)*x+ O(x^2) on page 135 of the Patterson book. The correct value of c(1) is A090998. - R. J. Mathar, Jul 11 2025

			0.65587807152025388107701951514539048127976638047858434729236244568387...

G. C. Greubel, Table of n, a(n) for n = 0..10000
R. J. Mathar, Erratum to Exercise A4.2 in "An Introduction to the Theory of the Riemann Zeta Function", viXra:2507.0094 (2025)
S. J. Patterson, An introduction to the theory of the Riemann zeta function, Cambridge studies in advanced mathematics no. 14, (1988) p. 135

R:= RealField(100); (Pi(R)^2 - 6*EulerGamma(R)^2)/12; // G. C. Greubel, Sep 05 2018

RealDigits[(Zeta[2] - EulerGamma^2)/2, 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)

PARI
```
-(Euler^2-zeta(2))/2
```

(EulerGamma^2 - zeta(2))/2 = -0.65587807152025388....

Equals A072691 - A155969/2.

A059190 Engel expansion of gamma^2, (gamma is the Euler-Mascheroni constant A001620) = 0.333178.

Original entry on oeis.org

4, 4, 4, 4, 4, 6, 23, 26, 126, 132, 154, 269, 421, 911, 1899, 7335, 14245, 34244, 78354, 173699, 239896, 247397, 659900, 1646344, 2454988, 6831657, 65833355, 839918922, 1187969748, 3583279448, 4114383765, 6590212761, 11304687651

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. A155969.

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 *)

A182497 Decimal expansion of gamma^3, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Volker Werner, May 02 2012

			0.19231551682118458966319237441963590712167826133337...

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

Cf. A001620, A155969.

R:= RealField(100); EulerGamma(R)^3; // G. C. Greubel, Sep 01 2018

Mathematica
```
RealDigits[EulerGamma^3, 10, 105][[1]]
```

default(realprecision, 100); Euler^3 \\ G. C. Greubel, Dec 26 2016

Equals A001620^3.

A229156 Decimal expansion of the negated value of the integral over (1/(1-y) + 1/log(y))*log(1-y)/y between 0 and 1.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 15 2013

			-0.91624014984429583053480927562573338...

G. C. Greubel, Table of n, a(n) for n = 0..2500
D. Zagier, A Kronecker limit formula for real quadratic fields, Mathem. Ann. 213 (2) (1975) 153-184, value of F(1), equation (7.12).

RealDigits[N[EulerGamma^2/2 + Pi^2/12 + StieltjesGamma[1], 2501]][[1]] (* G. C. Greubel, Dec 26 2016 *)

intnum(y=0, 1, (1/(1-y)+1/log(y)) *log(1-y) /y) \\ Michel Marcus, Dec 26 2016

Equals A155969/2 + A072691 + A082633.

A091558 Decimal expansion of zeta(Gamma^2).

Original entry on oeis.org

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

Offset: 0

Jon Perry, Mar 04 2004

			zeta(gamma^2)=-0.973022889025895614670184289515430656641758541138042827...

Cf. A155969, A251734.

Zeta(gamma^2) ; evalf(%) # R. J. Mathar, Aug 27 2024

RealDigits[Zeta[EulerGamma^2],10,120][[1]] (* Harvey P. Dale, Nov 09 2017 *)

PARI
```
zeta(Euler^2)
```

cp's OEIS Frontend

A073004 Decimal expansion of exp(gamma).

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A002389 Decimal expansion of -log(gamma), where gamma is Euler's constant A001620.

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A081855 Decimal expansion of Gamma''(1).

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A344964 Decimal expansion of the sum of the reciprocals of the squares of the zeros of the digamma function.

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

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A070860 Decimal expansion of Pi^2/12 - gamma^2 /2.

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A059190 Engel expansion of gamma^2, (gamma is the Euler-Mascheroni constant A001620) = 0.333178.

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