A080130 Decimal expansion of exp(-gamma).

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

Offset

0,1

Comments

By Mertens's third theorem, lim_{k->oo} (H_{k-1}*Product_{prime p<=k} (1-1/p)) = exp(-gamma), where H_n is the n-th harmonic number. Let F(x) = lim_{n->oo} ((Sum_{k<=n} 1/k^x)*(Product_{prime p<=n} (1-1/p^x))) for real x in the interval 0 < x < 1. Consider the function F(s) of the complex variable s, but without the analytic continuation of the zeta function, in the critical strip 0 < Re(s) < 1. - Thomas Ordowski, Jan 26 2023

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.5 p. 29, 2.7 p. 117 and 5.4 p. 285.

Links

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

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 202.

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628; arXiv:1303.1856 [math.NT], 2013.

Doron Zeilberger and Noam Zeilberger, Fractional Counting of Integer Partitions, 2018; Local copy [Pdf file only, no active links]

Formula

Equals lim inf_{n->oo} phi(n)*log(log(n))/n. - Arkadiusz Wesolowski, Aug 26 2012

From Alois P. Heinz, Dec 05 2018: (Start)

Equals lim_{n->oo} A322364(n)/(n*A322365(n)).

Equals lim_{n->oo} A322380(n)/A322381(n). (End)

Equals lim_{k->oo} log(k)*Product_{prime p<=k} (1-1/p). - Amiram Eldar, Jul 09 2020

Equals lim_{n->oo} A007838(n)/A000142(n). - Alois P. Heinz, Feb 24 2022

Equals Product_{k>=1} (1+1/k)*exp(-1/k). - Amiram Eldar, Mar 20 2022

Equals A001113^(-A001620). - Omar E. Pol, Dec 14 2022

Equals lim_{n->oo} (A001008(p_n-1)/A002805(p_n-1))*(A038110(n+1)/A060753(n+1)), where p_n = A000040(n). - Thomas Ordowski, Jan 26 2023

Example

0.56145948356688516982414321479088078676571...

Maple

evalf(exp(-gamma), 120);  # Alois P. Heinz, Feb 24 2022

Mathematica

RealDigits[N[Exp[-EulerGamma], 200]][[1]] (* Arkadiusz Wesolowski, Aug 26 2012 *)

Prog

(PARI) default(realprecision, 100); exp(-Euler) \\ G. C. Greubel, Aug 28 2018
(Magma) R:= RealField(100); Exp(-EulerGamma(R)); // G. C. Greubel, Aug 28 2018

Crossrefs

Cf. A000010, A000142, A001113, A001620, A002852, A007838, A073004, A079650, A322364, A322365, A322380, A322381.

Sequence in context: A298171 A046614 A371048 * A188640 A198419 A323738

Adjacent sequences: A080127 A080128 A080129 * A080131 A080132 A080133

Keyword

cons,nonn

Author

Benoit Cloitre, Jan 26 2003

Status

approved