A088541 -id:A088541 - cp's OEIS frontend

A088540 Decimal expansion of (4/sqrt(Pi))exp(-gamma/2)K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

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

Offset: 1

Benoit Cloitre, Nov 16 2003

An illustration of the Chebyshev effect.

			1.2923041571286886071...

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 100.

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See page 5.
S. Uchiyama, On some products involving primes, Proc. Amer. Math. Soc. 28 (1971) 629-630; MR 43#3227.

Cf. A087197, A064533, A088541, A155739.

digits = 105; LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/DirichletBeta[2^n])^(1/2^(n + 1)), {n, 1, 24}, WorkingPrecision -> digits + 5]; 4/Sqrt[Pi]*Exp[-EulerGamma/2]*LandauRamanujanK // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Jun 04 2014, updated Mar 14 2018 *)

Equals (4/sqrt(Pi))*exp(-gamma/2)*K = lim_{x->oo} Product_{p prime, p == 1 (mod 4), p <= x} (1 - 1/p).

Equals 4*A087197*A064533/exp(A155739). - R. J. Mathar, Feb 05 2009

Offset corrected by R. J. Mathar, Feb 05 2009

A125313 Decimal expansion of 2*exp(-gamma).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Dec 08 2006

			1.12291896713377033964828642958176157353142077385030633630831815209...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3, Landau-Ramanujan constant, p. 100.

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial Journal 1: 12-28, (1995) DOI:10.1080/03461238.1995.10413946.
Bernard Montaron, Exponential prime sequences, arXiv:2011.14653 [math.NT], 2020.
Simon Plouffe, Plouffe's Inverter.
S. K. Wilson and B. R. Duffy, An asymptotic analysis of small holes in thin fluid layers, Journal of Engineering Mathematics, July 1996, Volume 30, Issue 4, pp 445-457.

R:= RealField(100); 2*Exp(-EulerGamma(R)); // G. C. Greubel, Sep 05 2018

RealDigits[2*Exp[-EulerGamma], 10, 111][[1]]

default(realprecision, 100); 2*exp(-Euler) \\ G. C. Greubel, Sep 05 2018

Equals 2*A080130, 2*A001113^(-A001620) and 2/A073004 = 2/A068985^A001620.
Equals A088540 * A088541. - Jean-François Alcover, Jun 04 2014
Equals exp(A002162 - A001620). - John W. Nicholson, Apr 03 2015

A243377 Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 1 modulo 4} (1 + 1/p).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 04 2014

			0.732649819283832613620305823117683687363...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 101.

Eric Weisstein's World of Mathematics, Ramanujan constant.

Cf. A001620, A060294, A064533, A088541.

digits = 103; LandauRamanujanK =  1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/DirichletBeta[2^n])^(1/2^(n + 1)), {n, 1, 24}, WorkingPrecision -> digits + 5]; 4/Pi^(3/2)*Exp[EulerGamma/2]*LandauRamanujanK // RealDigits[#, 10, digits] & // First (* updated Mar 14 2018 *)

Equals (4/Pi^(3/2))*exp(gamma/2)*K, where gamma is the Euler-Mascheroni constant and K the Landau-Ramanujan constant.

Equals 2/(Pi*A088541) = A060294/A088541. - Amiram Eldar, Nov 16 2021

cp's OEIS Frontend

A088540 Decimal expansion of (4/sqrt(Pi))exp(-gamma/2)K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

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A125313 Decimal expansion of 2*exp(-gamma).

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A243377 Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 1 modulo 4} (1 + 1/p).

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cp's OEIS Frontend

A088540 Decimal expansion of (4/sqrt(Pi))*exp(-gamma/2)*K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

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Crossrefs

Programs

Mathematica

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Extensions

A125313 Decimal expansion of 2*exp(-gamma).

Views

Author

Keywords

Examples

References

Links

Programs

Magma

Mathematica

PARI

Formula

A243377 Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 1 modulo 4} (1 + 1/p).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A088540 Decimal expansion of (4/sqrt(Pi))exp(-gamma/2)K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.