A269330 -id:A269330 - cp's OEIS frontend

A270857 Decimal expansion of Sum_{n >= 1} G_n/n^2, where G_n are Gregory's coefficients.

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

Offset: 0

Iaroslav V. Blagouchine, Mar 24 2016

Gregory's coefficients (A002206 and A002207) are also known as (reciprocal) logarithmic numbers, Bernoulli numbers of the second kind and Cauchy numbers of the first kind. First few coefficients are G_1=+1/2, G_2=-1/12, G_3=+1/24, G_4=-19/720, etc.

			0.4826442216204626123794283911485757739701203962756656...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A270859, A269330, A001620, A002206, A002207, A195189.

evalf(int((Li(1+x)-gamma-ln(x))/x, x = 0..1), 120);

RealDigits[N[Integrate[(LogIntegral[1+x]-EulerGamma-Log[x])/x,{x,0,1}],150]][[1]]

Equals integral_{x=0..1} (li(1+x) - gamma - log(x))/x dx, where li(x) is the integral logarithm.

Mathematica program corrected by Harvey P. Dale, Jul 05 2022

A270859 Decimal expansion of Sum_{n >= 1} |G_n|/n^2, where G_n are Gregory's coefficients.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 24 2016

			0.5290529699404390240722939394755897280940381716959625...

Bernard Candelpergher, Ramanujan summation of divergent series, Berlin: Springer, 2017. See p. 105, eq. (3.23).

Iaroslav V. Blagouchine and Marc-Antoine Coppo, A note on some constants related to the zeta-function and their relationship with the Gregory coefficients, The Ramanujan Journal, Vol. 47 (2018), pp. 457-473. See p. 470, eq. (37); arXiv preprint, arXiv:1703.08601 [math.NT], 2017.
Mümün Can, Ayhan Dil, Levent Kargin, Mehmet Cenkci and Mutlu Güloglu, Generalizations of the Euler-Mascheroni constant associated with the hyperharmonic numbers, arXiv:2109.01515 [math.NT], 2021.

Cf. A001620, A002206, A002207, A013661, A082633, A131688, A195189, A269330, A270857.

evalf(int((-Li(1-x)+gamma+ln(x))/x, x = 0..1), 150)

N[Integrate[(-LogIntegral[1 - x] + EulerGamma + Log[x])/x, {x, 0, 1}], 150]

Equals Integral_{x=0..1} (-li(1-x) + gamma + log(x))/x dx, where li(x) is the logarithmic integral.

Equals A131688 + gamma_1 + gamma^2/2 - zeta(2)/2, where gamma_1 = A082633 and gamma = A001620 (Candelpergher, 2017; Blagouchine and Coppo, 2018). - Amiram Eldar, Mar 18 2024

cp's OEIS Frontend

A270857 Decimal expansion of Sum_{n >= 1} G_n/n^2, where G_n are Gregory's coefficients.

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