A088540 -id:A088540 - cp's OEIS frontend

A088541 Decimal expansion of sqrt(Pi)/(2K)*exp(-gamma/2) where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

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

Offset: 0

Benoit Cloitre, Nov 16 2003

An illustration of the Chebyshev effect.

			0.868927768234323...

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 6.
S. Uchiyama, On some products involving primes, Proc. Amer. Math. Soc. 28 (1971) 629-630; MR 43#3227.

Cf. A002161, A064533, A088540, A155739.

digits = 104; 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]; Sqrt[Pi]/(2*LandauRamanujanK )*Exp[-EulerGamma/2] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 04 2013, updated Mar 14 2018 *)

Equals sqrt(Pi)/(2K)*exp(-gamma/2) = lim x-->oo prod(1-1/p) where p runs through the primes p==3 mod 4 and p<=x.

Equals A002161*A064533/(2*exp(A155739)). - Michel Marcus, Jun 19 2020

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

A243378 Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 3 modulo 4} (1 + 1/p).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 04 2014

			0.985247581006096343690510604298896801...

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

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Ramanujan constant.

Cf. A001620, A064533, A088538, A088540.

digits = 102; 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]; 1/Sqrt[Pi]*Exp[EulerGamma/2]*1/LandauRamanujanK // RealDigits[#, 10, digits] & // First (* updated Mar 14 2018 *)

Equals (1/sqrt(Pi))*exp(gamma/2)*1/K, where gamma is the Euler-Mascheroni constant (A001620) and K the Landau-Ramanujan constant (A064533).

Equals 4/(Pi*A088540) = A088538/A088540. - Amiram Eldar, Nov 16 2021

A369106 Decimal expansion of sqrt(e^gamma/Pi).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jan 13 2024

			0.75294950604642059593549975758760481083...

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See Theorem 3.1 and Equation 14 at page 5.

Cf. A000796, A001113, A001620, A073004, A088540, A243376.

Flatten[First[RealDigits[Sqrt[E^EulerGamma/Pi],10,100]]]

cp's OEIS Frontend

A088541 Decimal expansion of sqrt(Pi)/(2K)*exp(-gamma/2) where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

References

Links

Programs

Magma

Mathematica

PARI

Formula

A243378 Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 3 modulo 4} (1 + 1/p).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A369106 Decimal expansion of sqrt(e^gamma/Pi).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica