A094640 -id:A094640 - cp's OEIS frontend

A256922 Decimal expansion of Sum_{k>=2} (-1)^kzeta(k)/(k2^k).

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			0.167825594815521207957737599259554003269226940067362331039...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights (2011) p. 272.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A001620, A002162, A053510, A094640, A256921.

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/2 + (1/2)*Log(Pi(R)) - Log(2); // G. C. Greubel, Sep 05 2018

RealDigits[EulerGamma/2 + (1/2)*Log[Pi] - Log[2], 10, 105] // First

Euler/2 + log(Pi)/2 - log(2) \\ Michel Marcus, Apr 13 2015

Equals A001620/2 + (1/2)*log(Pi) - log(2).
Equals Sum_{k>=1} (1/(2*k) - log(1 + 1/(2*k))). - Amiram Eldar, Jul 22 2020
Equals (A001620 - A094640)/2. - Ruud H.G. van Tol, Apr 26 2025

A126389 Numerators in a series for the "alternating Euler constant" log(4/Pi).

Original entry on oeis.org

1, -1, 2, -2, -1, 1, 1, -1, 1, -1, 3, -3, -2, 2, 2, -2, 2, -2, 2, -2, 4, -4, -3, 3, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, 3, -3, -1, 1, 1, -1, 1, -1, 3, -3, 1, -1, 3, -3, 3, -3, 5, -5, -4, 4, -2, 2, -2, 2, -2, 2, 2, -2, -2, 2, 2, -2, 2, -2, 2, -2, 4, -4, -2, 2

Offset: 2

Jonathan Sondow, Jan 01 2007

Nonzero values of (-1)^n*b(floor(n/2)) for n > 1, where b(n) = (# of 1's) - (# of 0's) in the base 2 expansion of n. The denominators of the series are A126388.

			floor(15/2) = 7 = 111 base 2, which has (# of 1's) - (# of 0's) = 3, so (-1)^15*3 = -3 is a term.

J. Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005) 61-65.
J. Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
Eric Weisstein's MathWorld, Digit Count

Cf. A037861, A066879, A094640, A094641, A110625, A110626, A126388.

b[n_] := DigitCount[n,2,1] - DigitCount[n,2,0]; L = {}; Do[If[b[Floor[n/2]] != 0, L = Append[L,(-1)^n*b[Floor[n/2]]]], {n,2,100}]; L

Log(4/Pi) = 1/2 - 1/3 + 2/6 - 2/7 - 1/8 + 1/9 + 1/10 - 1/11 + 1/12 - 1/13 + 3/14 - 3/15 - 2/16 + 2/17 + 2/22 - ...

A344716 Decimal expansion of (gamma + log(4/Pi))/2, where gamma is Euler's constant.

Original entry on oeis.org

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

Offset: 0

Kevin Ryde, May 27 2021

			0.40939007008601165264877449082284842...

Jean-Paul Allouche, Jeffrey Shallit, and Jonathan Sondow, Summation of Series Defined by Counting Blocks of Digits, Journal of Number Theory, volume 123, number 1, March 2007, pages 133-143. Also arXiv:math/0512399 [math.NT], 2005-2006.

Cf. A001620, A094640, A000120.

RealDigits[(EulerGamma + Log[4/Pi])/2, 10, 100][[1]] (* Amiram Eldar, May 27 2021 *)

Equals (A001620 + A094640)/2, the mean of Euler's constant and alternating Euler's constant.
Equals Sum_{n>=1} A000120(n) / (2*n*(2*n+1)), where A000120 is the number of 1-bits of n in binary. [Allouche, Shallit, Sondow]
Equals Sum_{k>=1} (1/(2*k-1) - log(1+1/(2*k-1))). - Amiram Eldar, Jun 19 2023

A374776 Decimal expansion of (2 + gamma - log(Pi/4))/4.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jul 19 2024

			0.7046950350430058263243872454114242138864661979...

Stephane Louboutin, Majorations explicites de |L(1, χ)| (Suite), C. R. Acad. Sci. Paris. 323, pp. 443-446 (1996). (In French). See Theorem 2 at p. 444.

Cf. A374775.

Cf. A000796, A001620, A094640, A256845.

RealDigits[(2+EulerGamma-Log[Pi/4])/4,10,100][[1]]

A307671 Decimal expansion of the alternating convergent series S = Sum_{k>=0} (-1)^kf(k), where f(k) = harmonic(2^k) - klog(2) - gamma, harmonic(m) is the Sum_{j=1..m} 1/j, and gamma is Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Luis H. Gallardo, Apr 20 2019

			0.272343587707596764784070676923955578748225108064395871645389620412837597...

Cf. A001620 (Euler-Mascheroni), A001008/A002805 (harmonic), A002162 (log(2)), A094640 (alternate Euler's constant), A256921 (a similar constant).

evalf(Sum((-1)^k*(harmonic(2^k) - k*log(2) - gamma), k=0..infinity), 120); # Vaclav Kotesovec, Apr 30 2019

digits = 104; s = NSum[(-1)^k*(HarmonicNumber[2^k] - k*Log[2] - EulerGamma), {k, 0, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+10]; RealDigits[s, 10, digits][[1]] (* Jean-François Alcover, Apr 28 2019 *)

default(realprecision, 120); sumalt(k=0, (-1)^k*(psi(2^k+1) - k*log(2))) \\ Vaclav Kotesovec, Apr 30 2019

cp's OEIS Frontend

A256922 Decimal expansion of Sum_{k>=2} (-1)^k*zeta(k)/(k*2^k).

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A126389 Numerators in a series for the "alternating Euler constant" log(4/Pi).

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A344716 Decimal expansion of (gamma + log(4/Pi))/2, where gamma is Euler's constant.

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A374776 Decimal expansion of (2 + gamma - log(Pi/4))/4.

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A307671 Decimal expansion of the alternating convergent series S = Sum_{k>=0} (-1)^k*f(k), where f(k) = harmonic(2^k) - k*log(2) - gamma, harmonic(m) is the Sum_{j=1..m} 1/j, and gamma is Euler-Mascheroni constant.

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A256922 Decimal expansion of Sum_{k>=2} (-1)^kzeta(k)/(k2^k).

A307671 Decimal expansion of the alternating convergent series S = Sum_{k>=0} (-1)^kf(k), where f(k) = harmonic(2^k) - klog(2) - gamma, harmonic(m) is the Sum_{j=1..m} 1/j, and gamma is Euler-Mascheroni constant.