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

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

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• changed

Comments

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)

Examples

			.549539312981644822337661768802907788330698981263...
		

References

• 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).

Links

• 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

Crossrefs

Cf. A001620, A073004, A155969, A213440.

Programs

Magma

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

Mathematica

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

PARI

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