A346589 -id:A346589 - cp's OEIS frontend

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

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.

A342934 Decimal expansion of gamma^(1/gamma), where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Christoph B. Kassir, Aug 30 2021

			0.385948254719841058037365008117537208453571562501405965467694054181966575...

Michael Ian Shamos, Shamos's catalog of the real numbers, (2011).

Cf. A001620, A098907, A345066, A346589, A073004, A182547, A182470, A073018.

Maple
```
Digits := 100; evalf(gamma^(1/gamma));
```

RealDigits[EulerGamme^(1/EulerGamma), 10, 100][[1]]

PARI
```
Euler^(1/Euler)
```

A346525 Decimal expansion of gamma/(1 - gamma), where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Jul 26 2021

Apart from the first digit the same as A091556. - R. J. Mathar, Aug 23 2021

			1.36527211862544155187721932845652201618831607176511...

Cf. A001620, A346589.

Cf. A002852 (continued fraction of gamma; omit first two terms for continued fraction of gamma/(1 - gamma)).

RealDigits[EulerGamma/(1 - EulerGamma), 10, 100][[1]] (* Alonso del Arte, Jul 26 2021 *)

PARI
```
Euler/(1-Euler)
```

A348362 Decimal expansion of gamma^gamma, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Christoph B. Kassir, Oct 14 2021

			0.7281831374925430411492169277934906172895559804281200678987723450693...

Michael Ian Shamos, Shamos's catalog of the real numbers, (2011).

Cf. A001620, A345066, A346589, A073004, A182547, A182470, A073018.

R:= RealField(100); EulerGamma(R)^EulerGamma(R);

Maple
```
Digits := 100; evalf(gamma^gamma);
```

RealDigits[EulerGamma^EulerGamma, 10, 100][[1]]

PARI
```
Euler^Euler
```

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

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A342934 Decimal expansion of gamma^(1/gamma), where gamma is the Euler-Mascheroni constant.

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Maple

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A346525 Decimal expansion of gamma/(1 - gamma), where gamma is the Euler-Mascheroni constant.

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Mathematica

PARI

A348362 Decimal expansion of gamma^gamma, where gamma is the Euler-Mascheroni constant.

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Magma

Maple

Mathematica

PARI