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

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, 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, 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

Offset: 1

Author

Ant King, Mar 11 2013

Keywords

• nonn
• cons

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.

Examples

			K3 = 8.066486182933632461051187438860461705800736710094589922443677...
		

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).

Crossrefs

Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222882.

Programs

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]

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

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.

Extensions

More terms from Robert G. Wilson v, Oct 19 2013