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A222882 Decimal expansion of Sierpiński's second constant, K2 = lim_{n->infinity} ((1/n) * (Sum_{i=1..n} A004018(i^2)) - 4/Pi * log(n)). 3
2, 2, 5, 4, 9, 2, 2, 4, 6, 2, 8, 8, 8, 2, 6, 4, 7, 6, 6, 2, 6, 8, 1, 8, 4, 7, 5, 9, 5, 2, 8, 7, 2, 3, 5, 5, 7, 8, 7, 1, 6, 6, 1, 5, 9, 8, 6, 0, 5, 3, 5, 1, 8, 8, 9, 1, 3, 8, 3, 1, 1, 6, 1, 8, 8, 5, 9, 1, 7, 2, 9, 2, 8, 9, 5, 9, 7, 1, 3, 9, 3, 4, 1, 0, 5, 8 (list; constant; graph; refs; listen; history; text; internal format)
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 second one, K2, and A222883 gives the decimal expansion of the third , K3. The formula given below show that K2 is related to several other, naturally occurring constants.

REFERENCES

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

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).

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

FORMULA

K2 = 4 / Pi * (eulergamma + K / Pi - 12 / Pi^2 * zeta'(2) + log(2) / 3 -1), where K  is Sierpiński's first constant (A062089) and eulergamma is the Euler-Mascheroni constant (A001620).

K2 = 4 * (12 * log(Gamma(3/4)) - 9*log(Pi) + 72*log(A) - 5*log(2) + 3 * eulergamma - 3) / (3 * Pi), where A is the Glaisher-Kinkelin constant (A074962).

K2 = 4 * (12 * log(Gamma(3/4)) + log(A^72 * e^(3*eulergamma - 3) / (32 * Pi^9))) / (3 * Pi).

K2 = 4 / Pi * (log(e^(3*eulergamma - 1) / (2^(2/3) * G^2)) - 12 / Pi^2 * zeta'(2)), where G is Gauss’ AGM constant (A014549).

K2 = 4 / Pi * (log(Pi^2 * e^(3*eulergamma - 1) / (2^(2/3) * L^2)) - 12 / Pi^2 * zeta'(2)), where L is Gauss’ lemniscate constant (A062539).

EXAMPLE

K2 = 2.25492246288826476626818475952872355787166159860535188913831...

MATHEMATICA

Take[Flatten[RealDigits[N[4(12 Log[Gamma[3/4]]-9 Log[Pi]+72 Log[Glaisher]-5 Log[2]+3 EulerGamma-3)/(3 Pi), 100]]], 86]

PROG

(PARI) 4/Pi*(log(exp(3*Euler-1)/(2^(2/3)/agm(sqrt(2), 1)^2)) - 12/Pi^2*zeta'(2)) \\ Charles R Greathouse IV, Dec 12 2013

CROSSREFS

Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222883.

Sequence in context: A318844 A034400 A021820 * A236335 A016724 A058657

Adjacent sequences:  A222879 A222880 A222881 * A222883 A222884 A222885

KEYWORD

nonn,cons

AUTHOR

Ant King, Mar 11 2013

EXTENSIONS

Minor edits by Vaclav Kotesovec, Nov 14 2014

STATUS

approved

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Last modified May 17 19:51 EDT 2022. Contains 353778 sequences. (Running on oeis4.)