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

1,2

Comments

An illustration of the Chebyshev effect.

References

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

Links

Table of n, a(n) for n=1..105.

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.

Formula

(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]

Example

1.2923041571286886071...

Mathematica

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 *)

Crossrefs

Cf. A087197, A064533, A088541, A155739.

Sequence in context: A272286 A019645 A011065 * A370117 A290102 A019826

Adjacent sequences: A088537 A088538 A088539 * A088541 A088542 A088543

Keyword

cons,nonn

Author

Benoit Cloitre, Nov 16 2003

Extensions

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

Status

approved