A086058 -id:A086058 - 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

A241011 Decimal expansion of Sierpiński's S~ (S "tilde" as named by S. Finch), a constant appearing in the asymptotics of the number of representations of a positive integer as a sum of two squares.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 07 2014

From Vaclav Kotesovec, Mar 10 2023: (Start)
Sum_{k=1..n} A002654(k)^2 ~ n * (log(n) + C) / 4, where C = A241011 =
4*gamma - 1 + log(2)/3 - 2*log(Pi) + 8*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.01662154573340811526279685971511542645018417752364748061...
The constant C, published by Ramanujan (1916, formula (22)), 4*gamma - 1 + log(2)/3 - log(Pi) + 4*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.3482276258576268... is wrong! (End)

			2.01662154573340811526279685971511542645018417752364748061...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's constant, p. 122.

Steven R. Finch, Errata and Addenda to Mathematical Constants, Section 2.10 p. 17.
Srinivasa Ramanujan, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, section (K), formula (22).

Cf. A062089, A002654, A086058, A241009.

S = 2*EulerGamma + 2*Log[2] + 3*Log[Pi] - 4* Log[Gamma[1/4]]; (* S~ *) St = 2*S - 12/Pi^2*Zeta'[2] + Log[2]/3 - 1; RealDigits[St, 10, 100] // First

S_tilde = 2*S - 12/Pi^2*zeta'(2) + log(2)/3 - 1, where S = A086058 - 1 = A062089 / Pi.

A241017 Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 08 2014

			0.822825249678847032995328716261464949475693118894850218393815613...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's Constant, p. 123.

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants, and other related results, arXiv:1701.07064 (2017) Proposition 9.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 103.
Guillaume Melquiond, W. Georg Nowak, Paul Zimmermann, Numerical approximation of the Masser-Gramain constant to four decimal places, Mathematics of Computation, Volume 82, Number 282, April 2013, Pages 1235-1246
Eric Weisstein's MathWorld, Sierpiński's Constant
Wikipedia, Sierpiński's Constant

Cf. A062089, A062539, A068466, A086058.

S = Log[4*Pi^3*Exp[2*EulerGamma]/Gamma[1/4]^4]; RealDigits[S, 10, 104] // First

log(agm(sqrt(2), 1)^2/2) + 2*Euler \\ Charles R Greathouse IV, Nov 26 2024

S = gamma + beta'(1) / beta(1), where beta is Dirichlet's beta function.
S = log(Pi^2*exp(2*gamma) / (2*L^2)), where L is Gauss' lemniscate constant.
S = log(4*Pi^3*exp(2*gamma) / Gamma(1/4)^4), where gamma is Euler's constant and Gamma is Euler's Gamma function.
S = A062089 / Pi, where A062089 is Sierpiński's K constant.
S = A086058 - 1, where A086058 is the conjectured (but erroneous!) value of Masser-Gramain 'delta' constant. [updated by Vaclav Kotesovec, Apr 27 2015]
S = 2*gamma + (4/Pi)*integral_{x>0} exp(-x)*log(x)/(1-exp(-2*x)) dx.
Sum_{k=1..n} r(k)/k = Pi*(log(n) + S) + O(n^(-1/2)).
Equals 2*A001620 - A088538*A115252 [Coffey]. - R. J. Mathar, Jan 15 2021

A241009 Decimal expansion of Sierpiński's S^ (Ŝ or "S hat" as named by S. Finch), a constant appearing in the asymptotics of the number of representations of a positive integer as a sum of two squares.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 07 2014

			1.7710119609560939428739802335360529080166503945687208610228709...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's constant, p. 122.

Steven R. Finch, Errata and Addenda to Mathematical Constants, Section 2.10 p. 17.

Cf. A062089, A086058.

S = 2* EulerGamma + 2*Log[2 ] + 3*Log[Pi] - 4* Log[Gamma[1/4]]; (* S^ *) Sh = EulerGamma + S - 12/Pi^2 Zeta'[2] + Log[2]/3 - 1; RealDigits[Sh, 10, 101] // First

3*Euler + 3*log(Pi) - 4*lngamma(1/4) - 12*zeta'(2)/Pi^2 + 7*log(2)/3 - 1 \\ Charles R Greathouse IV, Aug 08 2014

S_hat = gamma + S - 12/Pi^2*zeta'(2) + log(2)/3 - 1, where S = A086058 - 1 = A062089 / Pi.

A086057 Decimal expansion of a constant related to a conjectured value of the Masser-Gramain constant.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 07 2003

			0.646245439894813304266473396845792790...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 7.2, p. 459.

Eric Weisstein's World of Mathematics, Masser-Gramain Constant.

Cf. A086058.

RealDigits[ Pi/4*(2*EulerGamma + 2*Log[2] + 3*Log[Pi] - 4*Log[Gamma[1/4]]), 10, 102] // First (* Jean-François Alcover, Feb 07 2013, after Eric W. Weisstein *)

Equals Pi*(A086058 - 1)/4. - Jean-François Alcover, May 22 2014

Name corrected by Amiram Eldar, Jun 25 2023

cp's OEIS Frontend

A062089 Decimal expansion of Sierpiński's constant.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A241011 Decimal expansion of Sierpiński's S~ (S "tilde" as named by S. Finch), a constant appearing in the asymptotics of the number of representations of a positive integer as a sum of two squares.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A241017 Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A241009 Decimal expansion of Sierpiński's S^ (Ŝ or "S hat" as named by S. Finch), a constant appearing in the asymptotics of the number of representations of a positive integer as a sum of two squares.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A086057 Decimal expansion of a constant related to a conjectured value of the Masser-Gramain constant.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions