A340866 -id:A340866 - cp's OEIS frontend

A340839 Decimal expansion of Mertens constant C(5,1).

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

Offset: 1

Artur Jasinski, Jan 23 2021

Data taken from Alessandro Languasco and Alessandro Zaccagnini 2007.

			1.225238438539084580057609774749220527540595509391649938767...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens constants (pp. 94-95)

Vaclav Kotesovec, Table of n, a(n) for n = 1..499
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II: Numerical values, Math. Comp. 78 (2009), 315-326.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134 (digital data relative to the previous paper). [in this table on page 4, the last correct digit is a(109), beyond the level there certified. - _Vaclav Kotesovec_, Jan 26 2021]
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants mod q; 3 <= q <= 100, (2007) (GP-PARI procedure 100 digits accuracy).
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities., Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
For other links see A340711.

Cf. A077761, A083343, A091589, A138312, A161529, A230767, A238114, A271971, A340127, A340794, A340866.

A = C(5,1)=1.225238438539084580057609774749220527540595509391649938767...
B = C(5,2)=0.546975845411263480238301287430814037751996324100819295153...
C = C(5,3)=0.805951040448267864057376860278430932081288114939010897934...
D = C(5,4)=1.299364547914977988160840014964265909502574970408329662016...
A*B*C*D = 0.70182435445860646228... = (5/4)*exp(-gamma), where gamma is the Euler-Mascheroni constant A001620.
Formula from the article by Languasco and Zaccagnini, 2010, p.9:
A = ((13*sqrt(5)*Pi^2*exp(-gamma))/(150*log((1+sqrt(5))/2))*A340628/A340808)^(1/4).

Last 11 digits corrected by Vaclav Kotesovec, Jan 25 2021

More digits from Vaclav Kotesovec, Jan 26 2021

A335576 Decimal expansion of Mertens constant C(5,2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 26 2021

First 100 digits from Alessandro Languasco and Alessandro Zaccagnini 2007 p. 4.

			0.546975845411263480238301287430814...

Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134.

Cf. A001620, A336798, A340004, A340127, A340213, A340665, A340794, A340839, A340866.

A = C(5,1)=1.2252384385390845800576097747492205... see A340839.
B = C(5,2)=0.5469758454112634802383012874308140... this constant.
C = C(5,3)=0.8059510404482678640573768602784309... see A336798.
D = C(5,4)=1.2993645479149779881608400149642659... see A340866.
A*B*C*D = 0.70182435445860646228... = (5/4)*exp(-gamma), where gamma is the Euler-Mascheroni constant A001620.
B = sqrt(2)*5^(3/4)*sqrt(A340127)*exp(-gamma)/(4*sqrt(A340004)*A^2*C).
B = 2*A*D*log((1+sqrt(5))/2)/(C*sqrt(5)*A340794*A340665).
B = A*D*log((1+sqrt(5))/2)^2/(C*Pi*A340213^2).
From Vaclav Kotesovec, Jan 27 2021: (Start)
B*C = 5^(1/4) * exp(-gamma/2) * sqrt(log((1+sqrt(5))/2) / (2 * A340665 * A340794)).
A*D = 5^(3/4) * exp(-gamma/2) * sqrt(A340665 * A340794 / (8 * log((1+sqrt(5))/2))).
(End)

A336798 Decimal expansion of Mertens constant C(5,3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 27 2021

First 101 digits from Alessandro Languasco and Alessandro Zaccagnini 2007 p. 4.
A = C(5,1)=1.2252384385390845800576097747492205... see A340839.
B = C(5,2)=0.5469758454112634802383012874308140... see A335576.
C = C(5,3)=0.8059510404482678640573768602784309... this constant.
D = C(5,4)=1.2993645479149779881608400149642659... see A340866.

			0.80595104044826786405737686...

Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134.

Cf. A335576, A340839, A340866.

For formulas see A335576.

A340213 Decimal expansion of the constant kappa(-5) = (1/2)sqrt(sqrt(5)log(9+4sqrt(5))/(3Pi))sqrt(A340794A340665).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 26 2021

For general definition of the constants kappa(n) see Steven Finch 2009 p. 7, for this particular case kappa(-5) see p. 11.

			0.51593948227965348495312501394...

Steven Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2009 p. 7-11.

Cf. A340004, A340127, A340665, A340794, A340839, A340866.

Equals exp(-gamma/2)*log((1+sqrt(5))/2)*sqrt(5/Pi)/(2*C(5,2)*C(5,3)), where C(5,2) and C(5,3) are Mertens constants see A340839.
Equals 2*A340866*exp(gamma/4)*((1/5)*log((1+sqrt(5))/2))^(3/4)/sqrt(A340004).
Equals 2*A340866*exp(gamma/4)*log((1+sqrt(5))/2)/(sqrt(5*Pi)*A340884^(1/4)).
Equals 2*A340839*A340866*exp(gamma/2)*log((1+sqrt(5))/2)/sqrt(5*Pi).
Equals sqrt((1/3)*Pi*log(9+4*sqrt(5)))/(sqrt(5^(3/2)*A340004*A340127)). [Finch 2009 p. 11]

cp's OEIS Frontend

A340839 Decimal expansion of Mertens constant C(5,1).

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A335576 Decimal expansion of Mertens constant C(5,2).

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A336798 Decimal expansion of Mertens constant C(5,3).

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A340213 Decimal expansion of the constant kappa(-5) = (1/2)sqrt(sqrt(5)log(9+4sqrt(5))/(3Pi))sqrt(A340794A340665).

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cp's OEIS Frontend

A340839 Decimal expansion of Mertens constant C(5,1).

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A335576 Decimal expansion of Mertens constant C(5,2).

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A336798 Decimal expansion of Mertens constant C(5,3).

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A340213 Decimal expansion of the constant kappa(-5) = (1/2)*sqrt(sqrt(5)*log(9+4*sqrt(5))/(3*Pi))*sqrt(A340794*A340665).

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A340213 Decimal expansion of the constant kappa(-5) = (1/2)sqrt(sqrt(5)log(9+4sqrt(5))/(3Pi))sqrt(A340794A340665).