cp's OEIS Frontend

A222883 Decimal expansion of Sierpiński's third constant, K3 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} (A004018(i))^2) - 4* log(n)).

%I A222883 #40 Jun 28 2023 08:19:48
%S A222883 8,0,6,6,4,8,6,1,8,2,9,3,3,6,3,2,4,6,1,0,5,1,1,8,7,4,3,8,8,6,0,4,6,1,
%T A222883 7,0,5,8,0,0,7,3,6,7,1,0,0,9,4,5,8,9,9,2,2,4,4,3,6,7,7,1,3,3,7,9,1,2,
%U A222883 5,7,3,6,6,4,6,4,7,3,1,1,4,9,0,2,1,6,5,4,0,5,5,9,3,2,2,4,7,2,1,6,7,8,1,5,1
%N A222883 Decimal expansion of Sierpiński's third constant, K3 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} (A004018(i))^2) - 4* log(n)).
%C A222883 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.
%D A222883 Steven R. Finch, Mathematical Constants, Encyclopaedia of Mathematics and its Applications, Cambridge University Press (2003), p.123. Corrigenda in the link below.
%H A222883 G. C. Greubel, <a href="/A222883/b222883.txt">Table of n, a(n) for n = 1..10000</a>
%H A222883 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, (June 2012), pp. 15-16.
%H A222883 A. Schinzel, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa2112.pdf">Wacław Sierpiński’s papers on the theory of numbers</a>, Acta Arithmetica XXI, (1972), pp. 7-13. Corrigenda in "Dzieje Matematyki Polskiej" (Wrocław 2012), p.228 (in Polish).
%F A222883 K3 = 8*K / Pi - 48 / Pi^2 * zeta'(2) + 4 * log(2) / 3 - 4, where K  is Sierpinski's first constant (A062089).
%F A222883 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).
%F A222883 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).
%F A222883 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).
%F A222883 K3 = 4*K / Pi + Pi * K2 - 4 * eulergamma, where K2  is Sierpiński's second constant (A222882).
%F A222883 1 / 4 * K3 - 1 / 4 * Pi * K2 - log(pi^2 / (2 * L^2)) = eulergamma.
%F A222883 1 / 4 * K3 - 1 / 4 * Pi * K2 + log(2 * G^2) = eulergamma.
%e A222883 K3 = 8.066486182933632461051187438860461705800736710094589922443677...
%t A222883 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]
%o A222883 (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
%Y A222883 Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222882.
%K A222883 nonn,cons
%O A222883 1,1
%A A222883 _Ant King_, Mar 11 2013
%E A222883 More terms from _Robert G. Wilson v_, Oct 19 2013