A222882 -id:A222882 - cp's OEIS frontend

A062089 Decimal expansion of Sierpiński's constant.

2, 5, 8, 4, 9, 8, 1, 7, 5, 9, 5, 7, 9, 2, 5, 3, 2, 1, 7, 0, 6, 5, 8, 9, 3, 5, 8, 7, 3, 8, 3, 1, 7, 1, 1, 6, 0, 0, 8, 8, 0, 5, 1, 6, 5, 1, 8, 5, 2, 6, 3, 0, 9, 1, 7, 3, 2, 1, 5, 4, 4, 9, 8, 7, 9, 7, 1, 9, 3, 2, 0, 4, 4, 0, 0, 1, 1, 5, 7, 1, 2, 0, 2, 1, 1, 1, 1, 7, 7, 2, 4, 5, 2, 7, 0, 6, 4, 2, 8, 3, 0, 3, 1, 3, 4

Offset: 1

Jason Earls, Jun 27 2001

			2.5849817595792532170658935873831711600880516518526309173215...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 122-126.

Harry J. Smith, Table of n, a(n) for n = 1..5000
Steven R. Finch, Sierpinski's Constant. [Broken link]
Steven R. Finch, Sierpinski's Constant. [From the Wayback machine]
Simon Plouffe, Sierpinski Constant to 2000 digits. [Note: Last 2 digits given at this link are incorrect. - _William Echols_, Jun 04 2025]
Wacław Sierpiński, O sumowaniu szeregu Sigma_{n>a}^{n<=b} tau(n) f(n), gdzie tau(n) oznacza liczbę rozkładów liczby n na sumę kwadratów dwóch liczb całkowitych, Prace Matematyczno-Fizyczne, Vol. 18, No. 1 (1907), pp. 1-59.
Eric Weisstein's World of Mathematics, Sierpiński Constant.
Wikipedia, Sierpiński's constant.

Cf. A004018, A062083, A222882, A222883.

K=-Pi Log[Pi]+2 Pi EulerGamma+4 Pi Log[Gamma[3/4]];First@RealDigits[N[K,105]](* Ant King, Mar 02 2013 *)

-Pi*log(Pi)+2*Pi*Euler+4*Pi*log(gamma(3/4))

{ default(realprecision, 5080); x=-Pi*log(Pi)+2*Pi*Euler+4*Pi*log(gamma(3/4)); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062089.txt", n, " ", d)) } \\ Harry J. Smith, Aug 01 2009

Equals -Pi*log(Pi)+2*Pi*gamma+4*Pi*log(GAMMA(3/4)).
Equals Pi*A241017. - Eric W. Weisstein, Dec 10 2014
Equals Pi*(A086058-1). - Eric W. Weisstein, Dec 10 2014
Equals lim_{n->oo} (A004018(n)/n - Pi*log(n)). - Amiram Eldar, Apr 15 2021

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

Original entry on oeis.org

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

Ant King, Mar 11 2013

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.

			K3 = 8.066486182933632461051187438860461705800736710094589922443677...

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

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).

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

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]

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

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.

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

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A062089 Decimal expansion of Sierpiński's constant.

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

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