A016650 -id:A016650 - cp's OEIS frontend

A319232 Decimal expansion of Sum_{p = prime} 1/(p*log p)^2.

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

Offset: 0

R. J. Mathar, Sep 14 2018

Obtained by expanding the formalism of arXiv:0811.4739 to double integrals over the Riemann zeta function.

			1/A016627^2 + 1/A016650^2 + 1/8.047189^2 + ... = 0.637056184074676....

R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 (2008-2009).

Cf. A137245, A115563, A221711, A319231.

digits = 106; precision = digits + 10;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 300; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[(t - 2k) Log[Zeta[t]], {t, 2k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^3)*InLogZeta[k]]];
s = 0;
Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Jun 21 2022, after Vaclav Kotesovec *)

default(realprecision, 200); s=0; for(k=1, 300, s = s + moebius(k)/k^3 * intnum(x=2*k,[[1], 1], (x-2*k)*log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022

More terms from Vaclav Kotesovec, Jun 12 2022

A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.482037501770111223591657453125421381658405425375507779634198065524359698529473...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Hongwei Chen and G. C. Greubel, Sum of the Reciprocals of Polygonal Numbers (Solved), SIAM Problems and solutions.
Hongwei Chen and G. C. Greubel, Siam, Problems and Solutions, problem 07-003 and the solution

Cf. A000326, A016650, A093602, A294514.

Decimal expansion of the sum of the reciprocals of the m-gonal numbers: A000038 (m=3), A013661 (m=4), this sequence (m=5), A016627 (m=6), A244639 (m=7), A244645 (m=8), A244646 (m=9), A244647 (m=10), A244648 (m=11), A244649 (m=12), A275792 (m=14).

SetDefaultRealField(RealField(139)); R:= RealField(); 3*Log(3)-Pi(R)*Sqrt(3)/3; // G. C. Greubel, Mar 24 2024

RealDigits[Sum[2/(3*n^2-n), {n,1,Infinity}], 10, 111][[1]]
RealDigits[3*Log[3] - Pi*Sqrt[3]/3, 10, 140][[1]] (* G. C. Greubel, Mar 24 2024 *)

numerical_approx(3*log(3)-pi*sqrt(3)/3, digits=139) # G. C. Greubel, Mar 24 2024

Sum_{n>=1} 2/(3*n^2 - n).
Equals 3*log(3) - Pi*sqrt(3)/3 = A016650 - A093602. - Michel Marcus, Jul 03 2014
Equals 2*A294514. - Hugo Pfoertner, Apr 24 2025

A016455 Continued fraction for log(27).

Original entry on oeis.org

3, 3, 2, 1, 1, 1, 2, 2, 1, 4, 4, 1, 3, 10, 1, 4, 1, 5, 3, 5, 5, 1, 1, 1, 3, 108, 1, 7, 1, 1, 3, 3, 1, 1, 1, 8, 2, 1, 1, 1, 2, 3, 1, 3, 1, 7, 1, 1, 2, 12, 1, 50, 4, 3, 23, 3, 8, 3, 1, 5, 1, 4, 7, 1, 6, 1, 1, 5, 1, 1, 1, 1, 6, 15, 1, 2, 5, 2

Offset: 1

N. J. A. Sloane

			3.295836866004329074185735710... = 3 + 1/(3 + 1/(2 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, May 20 2009

Harry J. Smith, Table of n, a(n) for n = 1..20000

Cf. A016650 Decimal expansion. - Harry J. Smith, May 20 2009

ContinuedFraction[Log[27],80] (* Harvey P. Dale, May 02 2012 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(27)); for (n=1, 20000, write("b016455.txt", n, " ", x[n])); } \\ Harry J. Smith, May 20 2009

A382778 Decimal expansion of 6log(3)/(3log(3) - 3).

Original entry on oeis.org

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

Offset: 2

Stefano Spezia, May 11 2025

Upper bound for the irrationality measure of 3-adic analog of zeta(3) (see Lai et al., 2025 at p. 3).

			22.2814479514943215603962067415858532334689249078...

Li Lai, Johannes Sprang, and Wadim Zudilin, A note on the irrationality of zeta_2(5), arXiv:2505.05005 [math.NT], 2025.
Eric Weisstein's World of Mathematics, Irrationality Measure.
Wikipedia, Irrationality measure.

Cf. A002391, A016650, A382497, A383822, A383824.

RealDigits[6Log[3]/(3Log[3]-3),10,100][[1]]

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

Original entry on oeis.org

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

cp's OEIS Frontend

A319232 Decimal expansion of Sum_{p = prime} 1/(p*log p)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A016455 Continued fraction for log(27).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A382778 Decimal expansion of 6*log(3)/(3*log(3) - 3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A382778 Decimal expansion of 6log(3)/(3log(3) - 3).