A016627 -id:A016627 - cp's OEIS frontend

A280234 Decimal expansion of log(27)/log(27/4).

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

Offset: 1

Charles R Greathouse IV, Feb 24 2017

Appears as an exponent in an upper bound on the number of partitions of a set into disjoint unions; related to the ASTRAL algorithm in phylogenetic reconstruction.

			1.72598245787871910871908531940620853665960266205954942767875290916035...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Daniel Kane and Terence Tao, A bound on partitioning clusters, arXiv:1702.00912 [math.CO] (2017).
Daniel Kane and Terence Tao, A bound on partitioning clusters, Electronic Journal of Combinatorics, 24 (2) (2017), Article P2.31.
Index entries for transcendental numbers

Cf. A016627 (log(4)), A016650 (log(27)).

SetDefaultRealField(RealField(100)); Log(27)/Log(27/4); // G. C. Greubel, Oct 13 2018

RealDigits[N[Log[27]/(Log[27/4]), 100]] [[1]] (* Vincenzo Librandi, Feb 24 2017 *)

PARI
```
log(27)/log(27/4)
```

More digits from Jon E. Schoenfield, Mar 15 2018

A334388 Decimal expansion of Sum_{k>=1} A007953(k) / (k*(k+1)) where A007953(k) is the sum of digits of the integer k.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Sep 08 2020

This series is convergent.
Jeffrey Shallit generalizes this result to any base b (see Amer. Math. Month. link): Sum_{k>=1} digsum(k)_b / (k*(k+1)) = (b/(b-1)) * log(b) where digsum(k)_b is the sum of the digits of k when expressed in base b.
Sum_{n <= x} s(floor(x/n)) = kx + O(x^(2/3 + o(1))) where s(n) is the digital sum A007953 and k is this constant. See Bordellès, Dai, Heyman, Pan, & Shparlinski, Example 3.4. - Charles R Greathouse IV, Mar 22 2022

			2.5584278811044952044644349496492935640012238762541921955925865566

Jean-Paul Allouche, Somme de séries de nombres réels, Image des Mathématiques, CNRS, 2010 (in French).
Olivier Bordellès, Lixia Dai, Randell Heyman, Hao Pan, and Igor E. Shparlinski, On a sum involving the Euler function, arXiv:1808.00188 [math.NT]
J. O. Shallit, Solutions of Advanced Problems, 6450, The American Mathematical Monthly, Vol. 92, No. 7, Aug.-Sep., 1985, pp. 513-514; DOI: 10.2307/2322523.
Index entries for transcendental numbers

Cf. A002392 (log(10)), A007953 (digsum), A016627 (for base 2).

Cf. A308314.

Maple
```
evalf(10*log(10)/9,90);
```

RealDigits[10*Log[10]/9, 10, 100][[1]] (* Amiram Eldar, Sep 08 2020 *)

10*log(10)/9 \\ Charles R Greathouse IV, Mar 22 2022

Equals 1/(1*2) + 2/(2*3) + 3/(3*4) + 4/(4*5) + ... + 1/(10*11) + 2/(11*12) + ...

Equals (10/9) * log(10).

a(90) corrected by Georg Fischer, Jul 12 2021

A348373 Decimal expansion of Sum_{k>=1} H(k)^2/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 15 2021

			2.12538708076642786113951769297269016094950285280134...

István Mező, A q-Raabe formula and an integral of the fourth Jacobi theta function, Journal of Number Theory, Vol. 133, No. 2 (2013), pp. 692-704.
Seán Mark Stewart, Explicit evaluation of some quadratic Euler-type sums containing double-index harmonic numbers, Tatra Mountains Mathematical Publications, Vol. 77, No. 1 (2020), pp. 73-98.

Cf. A001008, A002805, A013661, A103930, A103931, A253191.

Similar constants: A016627, A076788.

RealDigits[Pi^2/6 + Log[2]^2, 10, 100][[1]]

Equals Pi^2/6 + log(2)^2 = A013661 + A253191.

A359532 Decimal expansion of 2*log(2)/Pi.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jan 04 2023

2*log(2)*n/Pi is also the dominant term in the asymptotic expansion of Sum_{k=1..n-1} (-1)^(k+1)*csc(Pi*k/n) at n tending to infinity. - Iaroslav V. Blagouchine, Apr 10 2025

			0.441271200305303186792912864235995381965...

Iaroslav V. Blagouchine, On a Generalization of Watson's Trigonometric Sum (On Dowker's Sum of Order One Half), INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 25, Article #A30, pp. 1-33, 2025. See p. 18.
John M. Campbell, Applications of a class of transformations of complex sequences, arXiv:2212.13305 [math.NT], 2022. See pp. 2, 8.

Cf. A000796, A001008, A002162, A002805, A002897, A016627, A016813, A060294.

Cf. A359533.

Mathematica
```
First[RealDigits[N[2Log[2]/Pi,98]]]
```

Equals A016627/A000796.
Equals A002162*A060294.
Equals 2*A284983.
Equals Sum_{i>=0} (-1/64)^i*binomial(2*i, i)^3*(4*i + 1)*H_{2*i}, where H_m is the m-th harmonic number (negated).

A387235 Decimal expansion of 2*log(2)/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 23 2025

Area enclosed by the curve of the equation x^6 + y^6 - x^3*y + x*y^3 = 0.

The asymptotic mean of A256232. - Amiram Eldar, Aug 23 2025

			0.46209812037329687294482141430545104538366675624...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 436.
Index entries for transcendental numbers.

Cf. A002162, A001504, A010701, A010722, A016627, A193535, A256232.

Mathematica
```
RealDigits[2Log[2]/3,10,100][[1]]
```

Equals A002162*A010722.
Equals log(4)/3 = A010701*A016627.
Equals Sum_{k>=0} (-1)^k/((3*k + 1)*(3*k + 2)) = Integral_{x=0..1} x^2*log(1 + 1/x^3) = -Integral_{x=0..1} log[1 - x^6]/x^4. [Shamos]
Equals A016627/3 = 2*A193535. - Hugo Pfoertner, Aug 23 2025

A196565 Decimal expansion of log(log(4)).

Original entry on oeis.org

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

Offset: 0

Kausthub Gudipati, Oct 04 2011

			0.3266342599782809824047929632255070986212366865231499910670022958228...

Cf. A002162, A016627, A074785, A194562, A074785.

RealDigits[Log[Log[4]],10,120][[1]] (* Harvey P. Dale, Aug 01 2021 *)

log(log(4)) \\ Charles R Greathouse IV, May 15 2019

From Amiram Eldar, Jun 12 2023: (Start)
Equals log(A016627).
Equals log(2) + log(log(2)) = A002162 - A074785. (End)

A241215 Decimal expansion of Sum_{n>=1} H(n)^4/(n+1)^3 where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 17 2014

			1.80161326804341290372948894202088843...

Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), page 16.

Cf. A016627, A083680, A102886, A152648, A152649, A152651, A233033, A233090, A238166, A238167, A239168, A238169, A238181, A238182, A238183, A240264.

37/180*Pi^4*Zeta[3] - 5/6*Pi^2*Zeta[5] - 109/8*Zeta[7] // RealDigits[#, 10, 100]& // First

37/2*zeta(3)*zeta(4) - 5*zeta(2)*zeta(5) - 109/8*zeta(7) \\ Stefano Spezia, Jan 19 2025

Equals (37/2)*zeta(3)*zeta(4) - 5*zeta(2)*zeta(5) - (109/8)*zeta(7).

Equals (37/180)*Pi^4*zeta(3) - (5/6)*Pi^2*zeta(5) - (109/8)*zeta(7).

A282821 Decimal expansion of Sum_{k >= 0} (4/(4k+1) - 3/(3k+1) + 2/(2*k+1) - 1/(k+1)).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Mar 03 2017

It is known that Sum_{k >= 0} Sum_{i = 1..h} (-1)^i*i/(i*k + 1) diverges for h = 3. This is the case h = 4, A016627 corresponds to the case h = 2.

			2.48171411447534970392757531472576785936281641070833471570388837547057328...

Cf. A016627.

RealDigits[(3 - Sqrt[3]) Pi/6 + Log[32] - Log[27]/2, 10, 100][[1]]

Equals (3 - sqrt(3))*Pi/6 + log(32) - log(27)/2.

A331239 Decimal expansion of Sum_{k>=0} (-1)^k/AGM(1, 1+k).

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Jan 13 2020

AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre.

This series is closely related to A188859 (Sum_{k>=0} (-1)^k/((1+(1+k))/2)) and A113024 (Sum_{k>=0} (-1)^k/sqrt(1+k)). The denominators of these alternating series differ by being arithmetic, geometric, or arithmetic-geometric means of 1 and k.

			0.6092151504524492287304733713491660511183939228565999735...

Cf. A016627, A188859, A113024.

PARI
```
sumalt(k=0, (-1)^k/agm(1,1+k))
```

A336518 Decimal expansion of 1 + Pi/log(4).

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Aug 10 2020

			3.26618007091359690481384147285833340508593073386189779209300827397030...

Cf. A000796, A016627.

Digits := 100: 1 + Pi/log(4): evalf(%, 100)*10^86:
ListTools:-Reverse(convert(floor(%), base, 10));

1 + Pi/log(4) \\ Michel Marcus, Aug 10 2020

Equals limit_{z->0} -z*Z(1-z, 1) - z*(Z(1-z, 1/4) - Z(1-z, 3/4))/(2^(1-z) - 2), where Z(v, w) is the Hurwitz zeta function.

Equals 1 + (PolyGamma(3/4) - PolyGamma(1/4)) / log(4).

cp's OEIS Frontend

A280234 Decimal expansion of log(27)/log(27/4).

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A334388 Decimal expansion of Sum_{k>=1} A007953(k) / (k*(k+1)) where A007953(k) is the sum of digits of the integer k.

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A348373 Decimal expansion of Sum_{k>=1} H(k)^2/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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A359532 Decimal expansion of 2*log(2)/Pi.

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A387235 Decimal expansion of 2*log(2)/3.

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A196565 Decimal expansion of log(log(4)).

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A241215 Decimal expansion of Sum_{n>=1} H(n)^4/(n+1)^3 where H(n) is the n-th harmonic number.

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A282821 Decimal expansion of Sum_{k >= 0} (4/(4k+1) - 3/(3k+1) + 2/(2*k+1) - 1/(k+1)).