A222883 - OEIS

%I #40 Jun 28 2023 08:19:48

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

%C 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 Steven R. Finch, Mathematical Constants, Encyclopaedia of Mathematics and its Applications, Cambridge University Press (2003), p.123. Corrigenda in the link below.

%H G. C. Greubel, <a href="/A222883/b222883.txt">Table of n, a(n) for n = 1..10000</a>

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, (June 2012), pp. 15-16.

%H 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 K3 = 8*K / Pi - 48 / Pi^2 * zeta'(2) + 4 * log(2) / 3 - 4, where K  is Sierpinski's first constant (A062089).

%F 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 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 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 K3 = 4*K / Pi + Pi * K2 - 4 * eulergamma, where K2  is Sierpiński's second constant (A222882).

%F 1 / 4 * K3 - 1 / 4 * Pi * K2 - log(pi^2 / (2 * L^2)) = eulergamma.

%F 1 / 4 * K3 - 1 / 4 * Pi * K2 + log(2 * G^2) = eulergamma.

%e K3 = 8.066486182933632461051187438860461705800736710094589922443677...

%t 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 (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 Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222882.

%K nonn,cons

%O 1,1

%A _Ant King_, Mar 11 2013

%E More terms from _Robert G. Wilson v_, Oct 19 2013