A363539 -id:A363539 - cp's OEIS frontend

A363538 Decimal expansion of Sum_{k>=1} (H(k) - log(k) - gamma)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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

Offset: 0

Amiram Eldar, Jun 09 2023

			0.72869391700393060593760589102029180041750271881292...

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in Pi^(-2) and into the formal enveloping series with rational coefficients only, Journal of Number Theory, Vol. 158 (2016), pp. 365-396.
Ovidiu Furdui, Problem 844, Problems and Solutions, The College Mathematics Journal, Vol. 38, No. 1 (2007), p. 61; Infinite sums and Euler's constant, Solution to Problem 844, ibid., Vol. 39, No. 1 (2008), pp. 71-72.

Cf. A001008, A001620, A002805, A072691, A082633, A363539, A363540.

RealDigits[-StieltjesGamma[1] - EulerGamma^2/2 + Pi^2/12, 10, 120][[1]]

Equals -gamma_1 - gamma^2/2 + Pi^2/12, where gamma_1 is the 1st Stieltjes constant (A082633).

A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 09 2023

			5.82174008504864652889686861550204134315033324319577...

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in Pi^(-2) and into the formal enveloping series with rational coefficients only, Journal of Number Theory, Vol. 158 (2016), pp. 365-396.

Cf. A001008, A001620, A002117, A013662, A082633, A086279, A086280, A363538, A363539.

RealDigits[-StieltjesGamma[3] - 3*EulerGamma*StieltjesGamma[2] - 3*EulerGamma^2*StieltjesGamma[1] - 3*EulerGamma^4/4 + 43*Zeta[4]/8, 10, 120][[1]]

Equals -gamma_3 - 3*gamma*gamma_2 - 3*gamma^2*gamma_1 - (3/4)*gamma^4 + (43/8)*zeta(4), where gamma_1, gamma_2 and gamma_3 are the 1st, 2nd and 3rd Stieltjes constants (A082633, A086279, A086280).

cp's OEIS Frontend

A363538 Decimal expansion of Sum_{k>=1} (H(k) - log(k) - gamma)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula