A019673 -id:A019673 - cp's OEIS frontend

A244647 Decimal expansion of the sum of the reciprocals of the decagonal numbers (A001107).

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

For the partial sums of the reciprocals of the (positive) decagonal numbers see A250551(n+1)/A294515(n), n >= 0. - Wolfdieter Lang, Nov 07 2017

			1.216745956158244182494339352004760382108361700922772890949837441544696356350....

Wikipedia, Polygonal number

Cf. A001107, A000038, A013661, A244639, A016627, A244645, A244646, A244648, A244649, A250551(n+1)/A294515(n).

RealDigits[ Log[2] + Pi/6, 10, 111][[1]] (* or *)
RealDigits[ Sum[1/(4n^2 - 3n), {n, 1 , Infinity}], 10, 111][[1]]

log(2)+Pi/6 \\ Charles R Greathouse IV, Feb 08 2023

Sum_{n>0} 1/(4n^2 - 3n) = log(2) + Pi/6, (A002162 + A019673).

A244854 Decimal expansion of Pi^2/32.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Jul 07 2014

Probability that a point selected uniformly at random from the unit 4-cube is in the unit 4-sphere.

Let S(n) = 1 - 1/3 + 1/5 - ... + ((-1)^(n-1))/(2n-1). Then Sum{n >=1} ((-1)^(n-1))*S(n) /(2n+1) = Pi^2 /32. The convergence is very slow. - Michel Lagneau, Feb 27 2015

			Choose -1 <= w, x, y, z <= 1 uniformly at random. Then this constant is the probability that w^2 + x^2 + y^2 + z^2 <= 1.

Index entries for transcendental numbers

Cf. A003881 (2-dimensional analog), A019673 (3-dimensional analog).

Digits:=100; evalf(Pi^2/32); # Wesley Ivan Hurt, Feb 27 2015

RealDigits[Pi^2/32,10,120][[1]] (* Harvey P. Dale, Jul 13 2014 *)

PARI
```
Pi^2/32
```

Equals Integral_{0..infinity} x^2*BesselK(0, x)^2 dx. - Jean-François Alcover, Apr 15 2015
Equals Integral_{x=0..1} arctan(x)/(1+x^2) dx. - Amiram Eldar, Aug 09 2020
Equals Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} (1 + x^2 + y^2 + z^2)^(-2). - Peter Luschny, Dec 10 2022

A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 03 2025

Polya (1954) defines the isoperimetric quotient of a solid as 36*Pi*V^2/(S^3), where V and S are the volume and surface area of the solid, respectively.

The isoperimetric quotient of a sphere is 1.

			0.30229989403903630843234637627369262204734437468212...

George Polya, Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics, Princeton University Press, Princeton, New Jersey, 1954. See pp. 188-189, exercise 43.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A273633 (sphericity).
Cf. isoperimetric quotient of other Platonic solids: A019673 (cube), A073010 (octahedron), A374772 (dodecahedron), A381672 (icosahedron).
Cf. A002194, A019673.

First[RealDigits[Pi/(6*Sqrt[3]), 10, 100]]

Equals Pi/(6*sqrt(3)) = A019673/A002194.

A019691 Decimal expansion of Pi/24.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

With a different offset, also decimal expansion of 5*Pi/12, 25*Pi/6 or 125*Pi/3. - Michel Marcus, Sep 09 2013

Volume of a quarter sphere of diameter 1. - Omar E. Pol, Aug 19 2019

			0.13089969389957471826927680763664595350821539164062940920728935801282...

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

Cf. A000796, A019670, A019673, A019675, A019679.

RealDigits[N[Pi/24,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

Pi/24 \\ Charles R Greathouse IV, Sep 09 2013

Equals A019673/4 or A019675/3 or A019679/2. - Omar E. Pol, Aug 19 2019

Equals (1/10) * Sum_{k>=1} sin(k*Pi/6)/k. - Amiram Eldar, May 30 2021

A247446 Decimal expansion of Pi*sqrt(3)/16.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Sep 29 2014

The atomic packing factor (APF) of the diamond cubic crystal lattice and the smallest APF among all crystallographic lattices filled by spheres of the same diameter. The APF of the body-centered-cubic (bcc) lattice packed with spheres of the same diameter is twice this value (see Examples). For other crystal lattices, see the cross-refs.

			0.340087380793915846986389673307904199803262421517388857919353425385273...
APF of the bcc lattice packed with spheres of the same diameter:
0.680174761587831693972779346615808399606524843034777715838706850770546...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Steve Sque, Structure of Diamond (adapted from 2005 thesis)
Wikipedia, Diamond cubic
Wikipedia, Atomic packing factor
Index entries for transcendental numbers

Cf. APF's of other crystal lattices: A093825 (hcp,fcc), A019673 (simple cubic).

RealDigits[Pi*Sqrt[3]/16,10,120][[1]] (* Vaclav Kotesovec, Oct 04 2014 *)

PARI
```
Pi*sqrt(3)/16
```

A273634 Decimal expansion of (Pi/6)^(1/3), the sphericity of the cube.

Original entry on oeis.org

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

Offset: 0

Felix Fröhlich, May 27 2016

			0.80599597700823482035848342331964246947230703616193077784614603...

Wikipedia, Sphericity.
Index entries for transcendental numbers.

Cf. A019673, A273633, A273635, A273636, A273637.

RealDigits[Surd[Pi/6, 3], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

default(realprecision, 50080); my(x=(Pi/6)^(1/3)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", "))

Equals cube root of A019673. - Michel Marcus, May 27 2016

A096615 Decimal expansion of 5 Pi^2/96.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jun 30 2004

			0.514041895...

Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See pp. 17-20.

Zafar Ahmed, Problem 10884, The American Mathematical Monthly, Vol. 108, No. 6 (2001), p. 566, Definitely an Integral, solution to Problem 10884, solved by Knut Dale, George L. Lamb, the proposer and others, ibid., Vol. 109, No. 7 (2002), pp. 670-671.
Zafar Ahmed, Ahmed's integral: the maiden solution, arXiv:1411.5169 [math.HO], 2014.
Michael Penn, Ahmed's Integral, YouTube video, 2021.
Juan Pla, A tale of Two Integrals: The Probability and Ahmed's Integrals, arXiv:1505.03314 [math.CA], 2015.
Eric Weisstein's World of Mathematics, Ahmed's Integral
Index entries for transcendental numbers

Cf. A019673, A098459, A102521, A244854.

RealDigits[5 Pi^2/96, 10 , 100][[1]] (* Amiram Eldar, Aug 17 2020 *)

From Amiram Eldar, Aug 17 2020: (Start)
Equals Integral_{x=0..1} arctan(sqrt(x^2 + 2))/(sqrt(x^2 + 2) * (x^2 + 1)) dx (Ahmed, 2001; Borwein et al., 2004).
Equals (1/10) * Integral_{x=1..oo} log(x)/(x^5 + x) dx. (End)

A228272 Volume of sphere (rounded down) with the diameter equal to n.

Original entry on oeis.org

0, 4, 14, 33, 65, 113, 179, 268, 381, 523, 696, 904, 1150, 1436, 1767, 2144, 2572, 3053, 3591, 4188, 4849, 5575, 6370, 7238, 8181, 9202, 10305, 11494, 12770, 14137, 15598, 17157, 18816, 20579, 22449, 24429, 26521, 28730, 31059, 33510, 36086, 38792, 41629, 44602

Offset: 1

K. D. Bajpai, Aug 19 2013

			a(6)=113 since volume is (Pi*n^3)/6 = Pi*6^3/6 = 113.0973355 and floor(113.0973355) = 113.

Georg Fischer, Table of n, a(n) for n = 1..5000 [first 578 terms from _K. D. Bajpai_]

Cf. A019673 (Pi/6).
Cf. A066645 (volume with radius n).
Cf. A228189 (similar sequence for right circular cone).

a:= n-> floor((Pi*n^3)/6):
seq(a(n),  n=1..100);

a(n) = floor((Pi*n^3)/6).

A338690 Inverse Moebius transform of A209615.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3

Offset: 1

Jianing Song, Apr 24 2021

Earliest occurrence of k is A018782(k).

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A209615, A035184 (a similar sequence), A018782, A002654, A019673.

f[p_, e_] := If[Mod[p, 4] == 1, e + 1, (1 + (-1)^e)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

a(n) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); if(p%4==1, r*=e+1, if(e%2, return(0)))); r

Multiplicative with a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p = 2 or p == 3 (mod 4).
a(n) = A002654(n) = A035184(n) for odd n. a(2^e * m) = a(m) for even m, 0 for odd m.
Dirichlet g.f.: zeta(s)*beta(s)/(1 + 2^(-s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/6 = 0.523598... (A019673). - Amiram Eldar, Oct 22 2022

A384513 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = 16/(n^2) + (1/n)i + 3/8 + (sqrt(3)/8)i to reach |z| > 2, starting with z = 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40

Offset: 1

Luke Bennet, May 31 2025

nonn

a(n)/n seems to converge to Pi/6.

a(n) counts the escape time of points outside the Mandelbrot set that converge to the Mandelbrot set's 1/6 period bulb. This is a proven fact and was the motivation for creating the sequence.

Luke Bennet, Table of n, a(n) for n = 1..10001
Thies Brockmöller, Oscar Scherz, and Nedim Srkalović, Pi in the Mandelbrot set everywhere, arXiv preprint arXiv:2505.07138 [math.DS], 2025.
Aaron Klebanoff, Pi in the Mandelbrot Set, Fractals 9 (2001), nr. 4, p. 393-402.

Cf. A019673, A097486, A383750, A384509.

def a(n):
    dps = 1
    while True:
        mpmath.iv.dps = dps
        c = iv.mpc(iv.mpf(16) / n ** 2 + 0.375, iv.mpf(1) / n + iv.sqrt(3) / 8)
        z = iv.mpc(0, 0)
        counter = 0
        while (z.real**2 + z.imag**2).b <= 4:
            z = z ** 2 + c
            counter += 1
        if (z.real**2 + z.imag**2).a > 4:
            return counter
        dps *= 2

cp's OEIS Frontend

A244647 Decimal expansion of the sum of the reciprocals of the decagonal numbers (A001107).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A244854 Decimal expansion of Pi^2/32.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A019691 Decimal expansion of Pi/24.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A247446 Decimal expansion of Pi*sqrt(3)/16.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A273634 Decimal expansion of (Pi/6)^(1/3), the sphericity of the cube.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A096615 Decimal expansion of 5 Pi^2/96.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A384513 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = 16/(n^2) + (1/n)i + 3/8 + (sqrt(3)/8)i to reach |z| > 2, starting with z = 0.