A097663 -id:A097663 - cp's OEIS frontend

A097669 Decimal expansion of the constant 5*exp(psi(3/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-5 linear recursions with varying coefficients (see A097680 for example).

			c = 1.90795953254354252255333813972952036908516068359082968228223...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097668, A097670-A097676.

RealDigits[ GoldenRatio^(Sqrt[5]/2)/5^(1/4)*E^(Pi/2Sqrt[1 - 2/Sqrt[5]]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
5*exp(psi(3/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(sqrt(5)/2)/5^(1/4)*exp(Pi/2*sqrt(1-2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

A097670 Decimal expansion of the constant 5*exp(psi(4/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-5 linear recursions with varying coefficients (see A097680 for example).

			c = 3.39276427892784980770475505655447128392740109258608442234780...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097669, A097671-A097676.

RealDigits[ GoldenRatio^(-Sqrt[5]/2)/5^(1/4)*E^(Pi/2Sqrt[1 + 2/Sqrt[5]]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
5*exp(psi(4/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(-sqrt(5)/2)/5^(1/4)*exp(Pi/2*sqrt(1+2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

A097672 Decimal expansion of the constant 6*exp(psi(5/6) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-6 linear recursions with varying coefficients (see A097681 for example).

			c = 4.38524598625080817369948994322956207765080128500902763042006...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097671, A097673-A097676.

RealDigits[1/Sqrt[12]*E^(Pi/2Sqrt[3]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
6*exp(psi(5/6)+Euler)
```

c = 1/sqrt(12)*exp(Pi/2*sqrt(3)).

More terms from Robert G. Wilson v, Aug 27 2004

A097674 Decimal expansion of the constant 8*exp(psi(3/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-8 linear recursions with varying coefficients (see A097682 for example).

			c = 0.90724581788216460753879452479208121378777525423587495906871...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097673, A097675-A097676.

RealDigits[(1 + Sqrt[2])^(Sqrt[2])/2E^(-Pi/2*(Sqrt[2] - 1)), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
8*exp(psi(3/8)+Euler)
```

c = (1+sqrt(2))^(sqrt(2))/2*exp(-Pi/2*(sqrt(2)-1)).

More terms from Robert G. Wilson v, Aug 27 2004

A097675 Decimal expansion of the constant 8*exp(psi(5/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-8 linear recursions with varying coefficients (see A097682 for example).

			c = 3.33325212658541721540039076972102211743980259727655469662827...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097674, A097676.

RealDigits[(1 + Sqrt[2])^(Sqrt[2])/2E^(Pi/2*(Sqrt[2] - 1)), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
8*exp(psi(5/8)+Euler)
```

c = (1+sqrt(2))^(sqrt(2))/2*exp(Pi/2*(sqrt(2)-1)).

More terms from Robert G. Wilson v, Aug 27 2004

A246499 Decimal expansion of zeta(2)/exp(gamma), gamma being the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 14 2014

It follows from Mertens theorem that this constant is the limit for large m of log(prime(m))*Product_{k=1..m} 1/(1 + 1/prime(k)).

			0.9235638316741813823235099539877039168469319632611163252035958316...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Mertens Theorem, Equations 5-9

Cf. A001113, A000796, A001620, A013661, A073004, A080130, A097663.

R:=RealField(100); Pi(R)^2/(6*Exp(EulerGamma(R))); // G. C. Greubel, Aug 30 2018

RealDigits[Zeta[2]/E^EulerGamma, 10, 100][[1]] (* Alonso del Arte, Nov 14 2014 *)

PARI
```
Pi^2/6/exp(Euler)
```

Equals Pi^2/(6*exp(gamma)).
Equals lim_{m->infinity} log(prime(m))*Product_{k=1..m} 1/(1 + 1/prime(k)).
Equals A013661/A073004. - Michel Marcus, Nov 18 2014

cp's OEIS Frontend

A097669 Decimal expansion of the constant 5*exp(psi(3/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097670 Decimal expansion of the constant 5*exp(psi(4/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097672 Decimal expansion of the constant 6*exp(psi(5/6) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097674 Decimal expansion of the constant 8*exp(psi(3/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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Mathematica

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Extensions

A097675 Decimal expansion of the constant 8*exp(psi(5/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A246499 Decimal expansion of zeta(2)/exp(gamma), gamma being the Euler-Mascheroni constant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs