A094644 -id:A094644 - 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

A079650 Continued fraction for e^(-gamma).

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 3, 5, 4, 1, 1, 2, 2, 1, 7, 9, 1, 16, 1, 1, 1, 2, 6, 1, 2, 1, 6, 2, 59, 1, 1, 1, 3, 3, 3, 2, 1, 3, 5, 100, 1, 58, 1, 2, 1, 94, 1, 1, 2, 2, 10, 1, 2, 7, 1, 3, 4, 5, 3, 10, 1, 21, 1, 11, 1, 4, 1, 2, 2, 1, 2, 2, 1, 8, 3, 2, 1, 1, 6, 1, 2, 2, 1, 38, 2, 1, 4, 1, 3, 1, 1, 5, 3, 1, 52, 1, 2, 2

Offset: 0

Neil Fernandez, Jan 22 2003

Same as A094644 except this number begins with a 0. - T. D. Noe, Jun 18 2012

			e^(-gamma) = 0.561... = 0 + 1/(1+ 1/ (1 +1/(3+...))), so sequence begins 0, 1, 1, 3,...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A080130 (exp(-gamma)), A094644.

ContinuedFraction[Exp[-EulerGamma], 100] (* Paolo Xausa, Aug 07 2024 *)

contfrac(exp(-Euler)) \\ Michel Marcus, Oct 13 2019

Offset changed by Andrew Howroyd, Aug 07 2024

A182551 Decimal expansion of gamma^(1/e), where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Volker Werner, May 04 2012

			0.81696075941989309813765514103027697644211120879263...

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

Cf. A001113, A001620, A073004, A094644.

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

RealDigits[ N[EulerGamma^(1/E), 105]][[1]]

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

Equals A001620^(1/A001113).

A064411 Increasing partial quotients of e^gamma.

Original entry on oeis.org

1, 3, 5, 7, 9, 16, 59, 100, 129, 314, 2294, 1568705

Offset: 1

Robert G. Wilson v, Sep 29 2001

Ronald L. Graham, D. E. Knuth and Oren Patashnik, "Concrete Mathematics, A Foundation for Computer Science," Addison-Wesley Publishing Co., Reading, MA, 1989, page 540.

Richard P. Brent, Computation of the regular continued fraction for Euler's constant, Mathematics of Computation 31 (1977), pages 771-777.

Cf. A073004 (e^gamma), A094644 (continued fraction).

t1 = ContinuedFraction[ E^EulerGamma, 10^5 ]; a = 0; Do[ If[ t1[ [ n ] ] > a, a = t1[ [ n ] ]; Print[ a ] ], {n, 1, 10^5} ]

A179950 Continued fraction for gamma^e, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

0, 4, 2, 4, 1, 14, 2, 1, 4, 1, 19, 3, 1, 4, 10, 13, 1, 5, 67, 3, 1, 3, 1, 1, 1, 1, 3, 11, 1, 1, 5, 4, 3, 3, 1, 16, 1, 1, 1, 2, 3, 5, 1, 1, 41, 1, 1, 4, 17, 3, 24, 1, 7, 307, 8, 1, 1, 2, 1, 2, 3, 2, 3, 4, 1, 2, 3, 1, 1, 21, 3, 17, 2, 5, 5, 1, 1, 2, 1, 2, 915, 1, 1, 2, 4, 1, 1, 1, 3, 1, 19, 54, 3, 1, 9, 8

Offset: 0

Michel Lagneau, Aug 03 2010

			gamma^e =.224517251983232062665128293743... = 0 + 1/(4 + 1/(2 + 1/(4 + 1/(1 + ...))))

Cf. A001620 (gamma), A073018 (decimal expansion).

Cf. A094644, A079650, A073004.

with(numtheory): Digits := 300: convert(evalf(gamma^exp(1)), confrac);

A322604 Factorial expansion of exp(gamma) = Sum_{n>=1} a(n)/n! with a(n) as large as possible.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 2, 4, 7, 5, 6, 5, 12, 1, 12, 9, 0, 7, 4, 14, 10, 17, 2, 14, 23, 4, 2, 2, 16, 2, 10, 18, 23, 26, 26, 26, 24, 1, 17, 26, 18, 12, 0, 15, 42, 34, 39, 33, 20, 18, 40, 43, 12, 47, 51, 10, 50, 35, 14, 23, 16, 1, 55, 41, 34, 29, 14, 41, 35, 60, 53, 45, 61, 35, 49, 73, 13, 13, 57, 59

Offset: 1

Tristan Cam, Dec 20 2018

nonn

Gamma is the Euler-Mascheroni constant (A001620).

			exp(gamma) = 1 + 1/2! + 1/3! + 2/4! + 3/5! + 4/6! + 2/7! + 4/8! + ...

Eric Weisstein's World of Mathematics, Harmonic Expansion
Index entries for factorial base representation

Cf. A073004 (decimal expansion), A094644 (continued fraction), A001620 (Euler-Mascheroni constant).

Digits:=200: a:=n->`if`(n=1,floor(exp(gamma)),floor(factorial(n)*exp(gamma))-n*floor(factorial(n-1)*exp(gamma))): seq(a(n),n=1..100); # Muniru A Asiru, Dec 20 2018

With[{b = Exp[EulerGamma]}, Table[If[n==1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = exp(Euler); for(n=1, 80, print1( if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

Sum_{n>=1} a(n)/n! = exp(gamma) = A073004.

A328339 Simple continued fraction expansion of Gamma(exp(gamma)+1).

Original entry on oeis.org

1, 1, 1, 1, 6, 15, 1, 4, 2, 1, 11, 2, 3, 1, 1, 13, 1, 1, 1, 1, 5, 1, 27, 1, 37, 1, 1, 2, 2, 2, 40, 1, 3, 1, 5, 1, 1, 3, 3942, 3, 2, 1, 13, 3, 11, 1, 2, 1, 15, 1, 1, 2, 1, 51, 37, 1, 1, 13, 4, 1, 1, 2, 5, 1, 1, 2, 1, 1, 5, 1, 75, 1, 16, 6, 2, 2, 1, 1, 7, 2, 4

Offset: 0

Daniel Hoyt, Oct 12 2019

'Gamma' is the gamma function, and 'gamma' is the Euler-Mascheroni constant.

Approximation of Gamma(exp(gamma)+1) in decimal: 1.6501566139104548059...

Cf. A094644, A073004, A079650.

ContinuedFraction[Gamma[1 + Exp[EulerGamma]], 81] (* Amiram Eldar, Oct 13 2019 *)

contfrac(gamma(exp(Euler)+1)) \\ Michel Marcus, Nov 04 2019

Gamma(exp(gamma)+1) = Gamma(exp(gamma)) * exp(gamma).

cp's OEIS Frontend

A073004 Decimal expansion of exp(gamma).

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A079650 Continued fraction for e^(-gamma).

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A182551 Decimal expansion of gamma^(1/e), where gamma is the Euler-Mascheroni constant.

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A064411 Increasing partial quotients of e^gamma.

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A179950 Continued fraction for gamma^e, where gamma is the Euler-Mascheroni constant.

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A322604 Factorial expansion of exp(gamma) = Sum_{n>=1} a(n)/n! with a(n) as large as possible.

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Mathematica

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A328339 Simple continued fraction expansion of Gamma(exp(gamma)+1).

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