A363538 -id:A363538 - cp's OEIS frontend

A363539 Decimal expansion of Sum_{k>=1} (H(k)^2 - (log(k) + gamma)^2)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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

Offset: 1

Amiram Eldar, Jun 09 2023

The formula for this sum was found by Olivier Oloa and proved by Roberto Tauraso in 2014.

			1.96896908391052854646489145379668054231137794286819...

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.
Olivier Oloa, A closed form of the series Sum_{n=1..oo} (H(n)^2 - (gamma + ln(n))^2)/n, Mathematics StackExchange, 2014.

Cf. A001008, A001620, A002117, A002805, A082633, A086279, A363538, A363540.

RealDigits[-StieltjesGamma[2] - 2*EulerGamma*StieltjesGamma[1] - 2*EulerGamma^3/3 + 5*Zeta[3]/3, 10, 120][[1]]

Equals -gamma_2 - 2*gamma*gamma_1 - (2/3)*gamma^3 + (5/3)*zeta(3), where gamma_1 and gamma_2 are the 1st and 2nd Stieltjes constants (A082633, A086279).

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

A363539 Decimal expansion of Sum_{k>=1} (H(k)^2 - (log(k) + gamma)^2)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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