A006752 -id:A006752 - cp's OEIS frontend

A378003 Decimal expansion of PiG - 7zeta(3)/4, where G = A006752.

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

Offset: 0

Stefano Spezia, Nov 14 2024

			0.773991201078871152328038383876510316276128388...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

Cf. A000796, A002117, A006752, A378021.

RealDigits[Pi Catalan-7Zeta[3]/4,10,100][[1]]

Pi*Catalan-7*zeta(3)/4 \\ Charles R Greathouse IV, Feb 12 2025

Equals Sum_{n>=1} (-1)^(n+1)/n^2 Sum_{k=0..n-1} 1/(2*k + 1) (see Finch).

A378021 Decimal expansion of PiG - 33zeta(3)/16, where G = A006752.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 14 2024

			0.3983484188414979381406202084041821941620701721...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

Cf. A000796, A002117, A006752, A378003.

RealDigits[Pi Catalan-33Zeta[3]/16,10,100][[1]]

Equals Sum_{n>=1} (-1)^(n+1)/n^2 Sum_{k=1..n} 1/(k + n) (see Finch).

A378910 Decimal expansion of 2*G/Pi, where G = A006752.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 10 2024

			0.58312180806163756027676891293678983772813230797167...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.10, p. 232.

Cf. A000796, A006752, A060294, A130834, A143233, A221209, A378911.

Mathematica
```
RealDigits[2Catalan/Pi,10,100][[1]]
```

Equals log(A130834).

Equals 2*A143233.

A378911 Decimal expansion of sqrt(2)exp(2G/Pi), where G = A006752.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Dec 10 2024

			2.5337372794858419095832896340418632916896308088420...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.10, p. 233.

Cf. A000796, A002193, A006752, A060294, A130834, A143233, A221209, A378910.

RealDigits[Sqrt[2]*Exp[2Catalan/Pi],10,100][[1]]

A383851 Decimal expansion of exp(8G/Pi)((1 - exp(-Pi/2))/(1 + exp(-Pi/2)))^2, where G is Catalan's constant (A006752).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, May 13 2025

			4.4312013071941991970823677283552872932838015281012...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Michael Penn, A beautiful Ramanujan product, YouTube video, 2025.

Cf. A006752, A049006, A377753.

First[RealDigits[Exp[8*Catalan/Pi]*((1 - #)/(1 + #))^2 & [Exp[-Pi/2]], 10, 100]]

Equals Product_{i=0..oo} (1 + 4/(2*i+1)^4)^((-1)^i*(2*i+1)) (from Ramanujan).

A345413 Decimal expansion of exp(gamma + M)(G - 7zeta(3)/(4*Pi))/4, where gamma is Euler's constant (A001620), M is Mertens's constant (A077761) and G is Catalan's constant (A006752).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 18 2021

This constant is notable for being the asymptotic limit in a formula derived by Sinha and Wolf (2010) which "brings together the elements from nine different topics of number theory" (see the Formula section).

			0.14248676756297667766013119038516485825699065019561...

Nilotpal Kanti Sinha and Marek Wolf, On a unified theory of numbers, arXiv:1009.4810 [math.NT], 2010-2011. See section 8, p. 11, eq. 37.

Cf. A001620, A002117, A006752, A077761.

Cf. A001008, A002805.

M = EulerGamma - NSum[PrimeZetaP[k]/k, {k, 2, Infinity}, WorkingPrecision -> 300, NSumTerms -> 300]; RealDigits[Exp[EulerGamma + M]*(Catalan - 7*Zeta[3]/(4*Pi))/4, 10, 100][[1]]

Equals lim_{n->oo} (1/log(n)^2) * Sum_{k=1..n} (1/gamma_k) * (1/k + 1/prime(k)) * (arctan(gamma_k/gamma_n))^2 * exp(H(k) + Sum_{i=1..k} 1/prime(i))), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and gamma_k is the imaginary part of the k-th nontrivial zero of the Riemann zeta function.

A377764 Decimal expansion of (Pi/8)exp(4G/Pi), where G is the Catalan constant (A006752).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 06 2024

			1.26052961282938641055453633013540984220266923935...

Michael Penn, A nice product from Ramanujan -- featuring 3 important constants!, YouTube video, 2023.

Cf. A000796, A001113, A005408, A006752, A016754, A019675, A229728.

First[RealDigits[Pi/8*Exp[4*Catalan/Pi], 10, 100]]

Equals A019675*A229728.
Equals Product_{k >= 1} (1 - 1/(2*k+1)^2)^((-1)^k*(2*k+1)) (from Ramanujan).
Equals Product_{k >= 1} (1 - 1/A016754(k))^((-1)^k*A005408(k)).

A377962 Decimal expansion of (8G - Pilog(2 + sqrt(3)))/3, where G = A006752.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 12 2024

			1.0634598331172279307675003455884827505711352959...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 100.

Cf. A000796, A006752, A065918.

RealDigits[(8Catalan-Pi Log[2+Sqrt[3]])/3,10,100][[1]]

Equals Sum_{k>=0} 1/(binomial(2*k,k)*(2*k + 1)^2) (see Finch and Shamos).

A377981 Decimal expansion of G/2 - Pi*log(2)/8, where G = A006752.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 13 2024

			0.18578453580065924121471564518649031196975172459...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 230.

Cf. A006752, A102886.

RealDigits[Catalan/2-Pi Log[2]/8,10,100][[1]]

Equals Sum_{n>=0} (-1)^(n+1)/(2*n + 1) Sum_{k=0..n-1} 1/(2*k + 1) (see Finch).
Equals Integral_{x=0..Pi/4} x*tan(x) dx (see Shamos).
Equals A006752/2 - A102886.

A378025 Decimal expansion of 1/2 - log(2)/4 - G/Pi, where G = A006752.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 14 2024

			0.035152300829194892507307513167060939117058812424...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 56.

Cf. A000796, A002162, A006752, A143233.

RealDigits[1/2-Log[2]/4-Catalan/Pi,10,100,-1][[1]]

Equals Sum_{n>=1} zeta(2*n)/(2^(4*n)*(2*n + 1)) (see Finch).

cp's OEIS Frontend

A378003 Decimal expansion of Pi*G - 7*zeta(3)/4, where G = A006752.

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A378021 Decimal expansion of Pi*G - 33*zeta(3)/16, where G = A006752.

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A378910 Decimal expansion of 2*G/Pi, where G = A006752.

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A378911 Decimal expansion of sqrt(2)*exp(2*G/Pi), where G = A006752.

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A383851 Decimal expansion of exp(8*G/Pi)*((1 - exp(-Pi/2))/(1 + exp(-Pi/2)))^2, where G is Catalan's constant (A006752).

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A345413 Decimal expansion of exp(gamma + M)*(G - 7*zeta(3)/(4*Pi))/4, where gamma is Euler's constant (A001620), M is Mertens's constant (A077761) and G is Catalan's constant (A006752).

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A377764 Decimal expansion of (Pi/8)*exp(4*G/Pi), where G is the Catalan constant (A006752).

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A377962 Decimal expansion of (8*G - Pi*log(2 + sqrt(3)))/3, where G = A006752.

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A378003 Decimal expansion of PiG - 7zeta(3)/4, where G = A006752.

A378021 Decimal expansion of PiG - 33zeta(3)/16, where G = A006752.

A378911 Decimal expansion of sqrt(2)exp(2G/Pi), where G = A006752.

A383851 Decimal expansion of exp(8G/Pi)((1 - exp(-Pi/2))/(1 + exp(-Pi/2)))^2, where G is Catalan's constant (A006752).

A345413 Decimal expansion of exp(gamma + M)(G - 7zeta(3)/(4*Pi))/4, where gamma is Euler's constant (A001620), M is Mertens's constant (A077761) and G is Catalan's constant (A006752).

A377764 Decimal expansion of (Pi/8)exp(4G/Pi), where G is the Catalan constant (A006752).

A377962 Decimal expansion of (8G - Pilog(2 + sqrt(3)))/3, where G = A006752.