A021031 -id:A021031 - cp's OEIS frontend

A020784 Decimal expansion of 1/sqrt(27).

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

Offset: 0

N. J. A. Sloane

This is the minimum ripple factor for a third-order Chebyshev filter for which the generalized reflectionless topology needs no negative elements. - Matthew A. Morgan, Oct 18 2017

			0.1924500897298752548363829268339858185492005837567089586728674....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.16, pp. 495, 527.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010482, A020760, A021031, A073010, A212886.

1/Sqrt(27); // G. C. Greubel, Feb 15 2018

RealDigits[1/Sqrt[27],10,200][[1]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)
realDigitsRecip[Sqrt[27]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jul 05 2025 *)

1/sqrt(27) \\ G. C. Greubel, Feb 15 2018

Equals Sum_{k>=0} binomial(2*k,k) * k/16^k. - Amiram Eldar, Aug 02 2020
Equals sqrt(3)/9. - Stefano Spezia, Dec 24 2024
Equals 1/A010482 = A020760/3 = sqrt(A021031) = A073010/Pi = A212886/2. - Hugo Pfoertner, Dec 24 2024

A343612 Decimal expansion of P_{3,2}(2) = Sum 1/p^2 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.30792075860773643684250507594099872658103266547551448005201925299378554901...

Jean-François Alcover, Table of n, a(n) for n = 0..1005
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A085548 (PrimeZeta(2)), A021031 (1/27).

Cf. A175644 (same for primes 3k+1), A086032 (for primes 4k+1), A085991 (for primes 4k+3), A343613 - A343619 (P_{3,2}(s): same with 1/p^s, s = 3, ..., 9).

digits = 105; nmax0 = 20; dnmax = 5;
Clear[PrimeZeta31];
PrimeZeta31[s_, nmax_] := PrimeZeta31[s, nmax] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2n + 1] ((1/(4n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, nmax}] // N[#, digits+5]&;
PrimeZeta31[2, nmax = nmax0];
PrimeZeta31[2, nmax += dnmax];
While[Abs[PrimeZeta31[2, nmax] - PrimeZeta31[2, nmax-dnmax]] > 10^-(digits+5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta32[2] = PrimeZetaP[2] - 1/3^2 - PrimeZeta31[2, nmax];
RealDigits[PrimeZeta32[2], 10, digits][[1]] (* Jean-François Alcover, May 06 2021, after M. F. Hasler's PARI code *)

s=0; forprimestep(p=2,1e8,3,s+=1./p^2);s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.
PrimeZeta32(s)={sumeulerrat(1/p^s)-1/3^s-suminf(n=0, my(t=s+2*n*s); moebius(2*n+1)*log((zeta(t)*(zetahurwitz(t, 1/6)-zetahurwitz(t, 5/6)))/((1+2^t)*(1+3^t)*zeta(2*t)))/(4*n+2))}
A343612_upto(N=100)={localprec(N+5); digits(PrimeZeta32(2)\.1^N)}

P_{3,2}(2) = P(2) - 1/3^2 - P_{3,1}(2) = A085548 - A000012 - A175644.

A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.134125178915463540428599329999431198995879919752168337370599106153853349956...

Jean-François Alcover, Table of n, a(n) for n = 0..1006
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=3) on p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A085541 (PrimeZeta(3)), A021031 (1/27).

Cf. A175645 (same for p==1 (mod 3)), A086033 (for primes 4k+1), A085992 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

s=0;forprimestep(p=2,1e8,3,s+=1./p^3);s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.

A343613_upto(N=100)={localprec(N+5); digits((PrimeZeta32(3)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32.

P_{3,2}(3) = P(3) - 1/3^3 - P_{3,1}(3) = A085541 - A021031 - A175645.

A021733 Decimal expansion of 1/729.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

729 = 3^6 = 9^3 = 27^2.
Period is 81 = 9^2 (see example for all 81 digits of the repeating part).
Repeating part in the form of 9 X 9 square table:
  1, 3, 7, 1, 7, 4, 2, 1, 1,
  2, 4, 8, 2, 8, 5, 3, 2, 2,
  3, 5, 9, 3, 9, 6, 4, 3, 3,
  4, 7, 0, 5, 0, 7, 5, 4, 4,
  5, 8, 1, 6, 1, 8, 6, 5, 5,
  6, 9, 2, 7, 2, 9, 7, 6, 6,
  8, 0, 3, 8, 4, 0, 8, 7, 7,
  9, 1, 4, 9, 5, 1, 9, 8, 9,
  0, 2, 6, 0, 6, 3, 1, 0, 0.
Note that each column consists of 9 consecutive (cyclically repeated) digits out of 10. The missing digits in columns from left to right are {7, 6, 5, 4, 3, 2, 0, 9, 8}, which form also a cycle of 9 out of 10 consecutive digits in reverse order, all digits except 1. - Alexander Adamchuk, Dec 28 2013

			1/729 = 0.00137174211248285322359396433470507544581618655692729766\
803840877914951989026063100 (period 81). - _Alexander Adamchuk_, Dec 28 2013

Cf. A068542 (period of the fraction 1/3^n).

Cf. A010701 (1/3), A000012 (1/9), A021031 (1/27), A021085 (1/81).

RealDigits[1/729, 10, 100] (* Alexander Adamchuk, Dec 28 2013 *)

PARI
```
1/729. \\ Michel Marcus, Oct 28 2019
```

Equals Sum_{k>=1} (k*(k+1)/2)/10^(k+2). - Davide Rotondo, Jun 11 2025

A214395 Decimal expansion of 16/27.

Original entry on oeis.org

5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5, 9, 2, 5

Offset: 0

Charles R Greathouse IV, Aug 07 2012

Betz's constant (or Betz's coefficient): maximum efficiency of a turbine. The constant is unfortunately named; Lanchester discovered it before Betz, and Lanchester cites Froude (though it seems that Lanchester was the first to write it explicitly). Joukowsky (also spelled Zhukovsky) and Munk published discoveries the same year as Betz.

			0.5925925925925925925925925925925925925925925925925925925925925925925...

Albert Betz, "Das Maximum der theoretisch möglichen Ausnutzung des Windes durch Windmotoren", Zeitschrift für das gesamte Turbinenwesen 26 (1920), pp. 307-309.
N. E. Joukowsky (Николай Егорович Жуковский), Windmill of the NEJ type (in Russian), Transactions of the Central Institute for Aero-Hydrodynamics of Moscow (1920). Appeared in Collected Papers Vol VI, The Joukowsky Institute for AeroHydrodynamics, Moscow, Russia (1937); VI: 405-409 (in Russian).
F. W. Lanchester, A contribution to the theory of propulsion and the screw propeller, Transactions of the Institution of Naval Architects 57 (1915), pp. 98-116. (See pp. 114-115.)

Albert Betz, The Maximum of the Theoretically Possible Exploitation of Wind by Means of a Wind Motor, Wind Engineering, Vol 37, Issue 4, 2013 (English translation).
Max M. Munk, Wind driven propeller, NACA Technical Memorandums 201 (1920), pp. 1-12 (German original in: Zeitschrift für Flugtechnik und Motorluftschiffahrt August 15, 1920: 220-223)
Henry Reich, Windmills Are NOT Like Dams, minutephysics video (2022).
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A021031, A021112.

RealDigits[16/27,10,120][[1]] (* or *) PadRight[{},120,{5,9,2}] (* Harvey P. Dale, Dec 04 2012 *)

PARI
```
16/27.
```

Equals 16 * A021031. - Felix Fröhlich, Jun 17 2017

cp's OEIS Frontend

A020784 Decimal expansion of 1/sqrt(27).

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A343612 Decimal expansion of P_{3,2}(2) = Sum 1/p^2 over primes == 2 (mod 3).

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A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).

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A021733 Decimal expansion of 1/729.

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A214395 Decimal expansion of 16/27.

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