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A222883 Decimal expansion of Sierpiński's third constant, K3 = lim_{n->infinity} ((1/n) * (Sum_{i=1..n} (A004018(i))^2) - 4* log(n)). 3
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 (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 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.

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

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

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

Sequence in context: A021996 A021128 A073235 * A091474 A059679 A198559

Adjacent sequences:  A222880 A222881 A222882 * A222884 A222885 A222886

KEYWORD

nonn,cons

AUTHOR

Ant King, Mar 11 2013

EXTENSIONS

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

STATUS

approved

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Last modified May 23 04:06 EDT 2022. Contains 353959 sequences. (Running on oeis4.)