

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
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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. 713. 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. 1516


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 GlaisherKinkelin constant (A074962) and eulergamma is the EulerMascheroni 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*Euler1)/(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



