A086058 -id:A086058

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A062089

Decimal expansion of Sierpiński's constant.

+10
11

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

REFERENCES

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

LINKS

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.

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.

FORMULA

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

EXAMPLE

2.5849817595792532170658935873831711600880516518526309173215...

MATHEMATICA

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

PROG

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

(PARI) { 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

CROSSREFS

Cf. A004018, A062083, A222882, A222883.

KEYWORD

cons,easy,nonn

AUTHOR

Jason Earls, Jun 27 2001

STATUS

approved

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.

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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)

REFERENCES

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

LINKS

Table of n, a(n) for n=1..100.

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).

FORMULA

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

EXAMPLE

2.01662154573340811526279685971511542645018417752364748061...

MATHEMATICA

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

CROSSREFS

Cf. A062089, A002654, A086058, A241009.

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Aug 07 2014

STATUS

approved

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.

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

REFERENCES

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

LINKS

Table of n, a(n) for n=0..103.

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

FORMULA

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

EXAMPLE

0.822825249678847032995328716261464949475693118894850218393815613...

MATHEMATICA

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

CROSSREFS

Cf. A062089, A062539, A068466, A086058.

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Aug 08 2014

STATUS

approved

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.

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

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

LINKS

Table of n, a(n) for n=1..101.

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

FORMULA

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

EXAMPLE

1.7710119609560939428739802335360529080166503945687208610228709...

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

Cf. A062089, A086058.

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Aug 07 2014

STATUS

approved

A086057

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

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..101.

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

FORMULA

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

EXAMPLE

0.646245439894813304266473396845792790...

MATHEMATICA

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 *)

CROSSREFS

Cf. A086058.

KEYWORD

nonn,cons,easy

AUTHOR

Eric W. Weisstein, Jul 07 2003

EXTENSIONS

Name corrected by Amiram Eldar, Jun 25 2023

STATUS

approved

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