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.

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

1,1

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.

Sequence in context: A278075 A114329 A361956 * A220905 A353451 A226573

Adjacent sequences: A241008 A241009 A241010 * A241012 A241013 A241014

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 07 2014

Status

approved