A243340 Decimal expansion of 4*L/(3*Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.
1, 1, 1, 2, 8, 3, 5, 7, 8, 8, 8, 9, 8, 7, 6, 4, 2, 4, 8, 3, 7, 5, 2, 3, 9, 6, 4, 3, 7, 3, 2, 0, 6, 2, 4, 1, 1, 9, 9, 1, 9, 9, 0, 6, 8, 4, 6, 5, 3, 7, 9, 6, 0, 0, 3, 2, 6, 6, 4, 3, 6, 4, 9, 3, 4, 7, 1, 5, 7, 5, 9, 9, 0, 2, 7, 9, 3, 6, 8, 5, 4, 9, 1, 5, 9, 5, 8, 8, 2, 1, 3, 8, 0, 1, 7, 0, 0, 4, 3, 2, 1, 7, 2, 0, 9
Offset: 1
Examples
1.11283578889876424837523964373206241199199...
References
- B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 102.
Links
- Eric Weisstein's World of Mathematics, Lemniscate Constant.
Programs
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Mathematica
L = Pi^(3/2)/(Sqrt[2]*Gamma[3/4]^2); RealDigits[4*L/(3*Pi), 10, 103] // First
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PARI
2*sqrt(2*Pi)/(3*gamma(3/4)^2) \\ Stefano Spezia, Nov 27 2024
Formula
Equals 2*sqrt(2*Pi)/(3*Gamma(3/4)^2).
From Peter Bala, Mar 24 2024: (Start)
An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 3 for k >= 0.
For example, taking k = 0 and k = 1 yields
4*L/(3*Pi) = 1 + 1/(6 + (5*7)/(6 + (9*11)/(6 + (13*15)/(6 + ... + (4*n + 1)*(4*n + 3)/(6 + ... ))))) and
4*L/(3*Pi) = 8/(7 + (1*3)/(14 + (5*7)/(14 + (9*11)/(14 + (13*15)/(14 + ... + (4*n + 1)*(4*n + 3)/(14 + ... )))))).
Equals (2/3) * 1/A076390. (End)