OFFSET
1,1
COMMENTS
Sierpiński introduced three constants in his 1908 doctoral thesis. The first, K, is very well known, bears his name and its decimal expansion is given in A062089. However, the second and third of these constants appear to have been largely forgotten. This sequence gives the decimal expansion of the third one, K3, and A222882 gives the decimal expansion of the second one, K2. The formula given below show that K3 is related to several other, naturally occurring constants including K and K2.
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopaedia of Mathematics and its Applications, Cambridge University Press (2003), p.123. Corrigenda in the link below.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, (June 2012), pp. 15-16.
A. Schinzel, Wacław Sierpiński’s papers on the theory of numbers, Acta Arithmetica XXI, (1972), pp. 7-13. Corrigenda in "Dzieje Matematyki Polskiej" (Wrocław 2012), p.228 (in Polish).
FORMULA
K3 = 8*K / Pi - 48 / Pi^2 * zeta'(2) + 4 * log(2) / 3 - 4, where K is Sierpinski's first constant (A062089).
K3 = 4 / 3 * log(A^72 * e^(6 * eulergamma - 3)*( Gamma(3/4))^24 / (32 * pi^12)), where A is the Glaisher-Kinkelin constant (A074962) and eulergamma is the Euler-Mascheroni constant (A001620).
K3 = 4*log(exp(5*eulergamma - 1) / (2^(5 / 3) * G^4)) - 48 / Pi^2 * zeta'(2) - 4* eulergamma, where G is Gauss’ AGM constant (A014549).
K3 = 4*log(Pi^4 * e^(5*eulergamma - 1) / (2^(5 / 3) * L^4)) - 48 / Pi^2 * zeta'(2) - 4* eulergamma, where L is Gauss’ lemniscate constant (A062539).
K3 = 4*K / Pi + Pi * K2 - 4 * eulergamma, where K2 is Sierpiński's second constant (A222882).
1 / 4 * K3 - 1 / 4 * Pi * K2 - log(pi^2 / (2 * L^2)) = eulergamma.
1 / 4 * K3 - 1 / 4 * Pi * K2 + log(2 * G^2) = eulergamma.
EXAMPLE
K3 = 8.066486182933632461051187438860461705800736710094589922443677...
MATHEMATICA
Take[RealDigits[N[4/3 (24*Log[Gamma[3/4]] - 12*Log[Pi] + 72*Log[Glaisher] - 5*Log[2] + 6*EulerGamma - 3), 100]][[1]], 86]
PROG
(PARI) 4*log(exp(5*Euler-1)/(2^(5/3)/agm(sqrt(2), 1)^4))-48/Pi^2*zeta'(2) - 4*Euler \\ Charles R Greathouse IV, Dec 12 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Ant King, Mar 11 2013
EXTENSIONS
More terms from Robert G. Wilson v, Oct 19 2013
STATUS
approved