A381671 -id:A381671 - cp's OEIS frontend

A093766 Decimal expansion of Pi/(2*sqrt(3)).

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

Offset: 0

Eric W. Weisstein, Apr 15 2004

Density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).
The number gives the areal coverage (90.68... percent) of the close hexagonal (densest) packing of circles in the plane. The hexagonal unit cell is a rhombus of side length 1 and height sqrt(3)/2; the area of the unit cell is sqrt(3)/2 and the four parts of circles add to an area of one circle of radius 1/2, which is Pi/4. - R. J. Mathar, Nov 22 2011
Ratio of surface area of a sphere to the regular octahedron whose edge equals the diameter of the sphere. - Omar E. Pol, Dec 09 2013

			0.906899682117108925297039128821077866142033124046370287784942...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.7, p. 506.
L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (84) on page 16.
Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 30.

J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discr. Comp. Geom., 13 (1995), 383-403.
Xi Lin, Dirk Schmelter, Sadaf Imanian, and Horst Hintze-Bruening, Hierarchically Ordered alpha-Zirconium Phosphate Platelets in Aqueous Phase with Empty Liquid, Scientific Reports (2019) Vol. 9, Article No. 16389.
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015. See Table 22 for L(m=6,r=2,s=1).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 625-626.
N. J. A. Sloane and Andrey Zabolotskiy, Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural).
Eckard Specht, May 21 2012, The best known packings of equal circles in a circle (complete up to N=1500).
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See p. 146.
Eric Weisstein's World of Mathematics, Smoothed Octagon.
Eric Weisstein's World of Mathematics, Circle Packing.
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646, (1/2)*A093602, A346585, A346584, A346583.

Cf. A072691, A381671.

RealDigits[Pi/(2 Sqrt[3]), 10, 111][[1]] (* Robert G. Wilson v, Nov 07 2012 *)

Pi/sqrt(12) \\ Charles R Greathouse IV, Oct 31 2014

Equals (5/6)*(7/6)*(11/12)*(13/12)*(17/18)*(19/18)*(23/24)*(29/30)*(31/30)*..., where the numerators are primes > 3 and the denominators are the nearest multiples of 6.
Equals Sum_{n>=1} 1/A134667(n). [Jolley]
Equals Sum_{n>=0} (-1)^n/A124647(n). [Jolley eq. 273]
Equals A000796 / A010469. - Omar E. Pol, Dec 09 2013
Continued fraction expansion: 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). See A254381 for a sketch proof. - Peter Bala, Feb 04 2015
From Peter Bala, Feb 16 2015: (Start)
Equals 4*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 5)).
Continued fraction: 1/(1 + 1^2/(4 + 5^2/(2 + 7^2/(4 + 11^2/(2 + ... + (6*n + 1)^2/(4 + (6*n + 5)^2/(2 + ... ))))))). (End)
The inverse is (2*sqrt(3))/Pi = Product_{n >= 1} 1 + (1 - 1/(4*n))/(4*n*(9*n^2 - 9*n + 2)) = (35/32) * (1287/1280) * (8075/8064) * (5635/5632) * (72819/72800) * ... =  1.102657790843585... - Dimitris Valianatos, Aug 31 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 3) dx.
Equals Integral_{x=0..oo} 1/(3*x^2 + 1) dx. (End)
Equals 1 + Sum_{k>=1} ( 1/(6*k+1) - 1/(6*k-1) ). - Sean A. Irvine, Jul 24 2021
For positive integer k, Pi/(2*sqrt(3)) = Sum_{n >= 0} (6*k + 4)/((6*n + 1)*(6*n + 6*k + 5)) - Sum_{n = 0..k-1} 1/(6*n + 5). - Peter Bala, Jul 10 2024
From Stefano Spezia, Jun 05 2025: (Start)
Equals Sum_{k>=0} (-1)^k/((k + 1)*(3*k + 1)).
Equals Integral_{x=0..oo} 1/(x^4 + x^2 + 1) dx.
Equals Integral_{x=0..oo} x^2/(x^4 + x^2 + 1) dx. (End)
Equals sqrt(A072691) = 3*A381671. - Hugo Pfoertner, Jun 05 2025

Entry revised by N. J. A. Sloane, Feb 10 2013

A374772 Decimal expansion of the upper bound of the density of sphere packing in the Euclidean 3-space resulting from the dodecahedral conjecture.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 19 2024

See A374753 for more information on the dodecahedral conjecture.

Also isoperimetric quotient (see A381671 for definition) of a regular dodecahedron. - Paolo Xausa, May 19 2025

			0.7546973993374058303916521059902293313424321921459...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Local Density.
Index entries for transcendental numbers.

Cf. A374753 (dodecahedral conjecture), A374755 (strong dodecahedral conjecture), A374771, A374837, A374838.

Cf. A093825, A019699, A102769, A131595, A381671.

First[RealDigits[Pi*Sqrt[5 + Sqrt[5]]/(15*Sqrt[10]*(Sqrt[5] - 2)), 10, 100]]

Pi*sqrt(5 + sqrt(5))/(15*sqrt(10)*(sqrt(5) - 2)) \\ Charles R Greathouse IV, Feb 07 2025

Equals (4/3)*Pi/A374753 = 10*A019699/A374753.
Equals A374771/A102769.
Equals Pi*sqrt(5 + sqrt(5))/(15*sqrt(10)*(sqrt(5) - 2)).
Equals 4*Pi/A374755.
Equals 36*Pi*A102769^2/(A131595^3). - Paolo Xausa, May 19 2025

A306771 Numbers m such that m = i + j = i * k and phi(m) = phi(i) + phi(j) = phi(i) * phi(k) for some i, j, k, where phi is the Euler totient function A000010.

Original entry on oeis.org

3, 15, 21, 33, 39, 51, 57, 69, 75, 87, 93, 105, 111, 123, 129, 141, 147, 159, 165, 177, 183, 195, 201, 213, 219, 231, 237, 249, 255, 267, 273, 285, 291, 303, 309, 321, 327, 339, 345, 357, 363, 375, 381, 393, 399, 411, 417, 429, 435, 447, 453, 465, 471, 483, 489

Offset: 1

Michel Lagneau, Mar 09 2019

The 55 terms given in the data section are consistent with a definition "numbers congruent to 3 or 15 mod 18". - Peter Munn, May 12 2020
The observation above is true for the first 10^4 terms. - Amiram Eldar, Dec 08 2020
The observation above is true for every term; see link. - Flávio V. Fernandes, Apr 18 2022
A001748 \ {6, 9} is a subsequence because, for p prime >= 5, 3 * p = p + 2p = p * 3 and phi(3p) = phi(p) + phi(2p) = phi(p) * phi(3) = 2 * (p-1). - Bernard Schott, May 13 2022

			33 is in the sequence because:
phi(33) = phi(11 + 22) = phi(11) + phi(22) = 10 + 10 = 20, and
phi(33) = phi(3 * 11) = phi(3) * phi(11) = 2 * 10 = 20.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Flávio V. Fernandes, A306771(n) equals 3 times A007310(n)
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000010, A000040, A007310, A001748, A381671.

with(numtheory):
for n from 1 to 500 do:
ii:=0:
  for i from 1 to trunc(n/2) while(ii=0) do:
   if phi(i)+ phi(n-i)= phi(n) and n/i = floor(n/i)
      and phi(i)*phi(n/i)=phi(n)
      then
      ii:=1:printf(`%d, `,n):
      else
   fi:
  od:
od:

LinearRecurrence[{1, 1, -1}, {3, 15, 21}, 100] (* Paolo Xausa, Mar 07 2025 *)

isok(m) = {my(phim = eulerphi(m)); for (i=1, m\2, if ((eulerphi(i) + eulerphi(m-i) == phim) && !frac(m/i) && (eulerphi(m/i)*eulerphi(i) == phim), return (1)););} \\ Michel Marcus, Mar 09 2019

From Chai Wah Wu, Mar 07 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
G.f.: x*(3*x^2 + 12*x + 3)/((x - 1)^2*(x + 1)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) (A381671). - Amiram Eldar, Mar 08 2025

Incorrect comment deleted by Peter Munn, May 12 2020

Name corrected by Flávio V. Fernandes, Aug 26 2021 and Peter Munn, Sep 03 2021

A381672 Decimal expansion of the isoperimetric quotient of a regular icosahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 03 2025

For the definition of isoperimetric quotient of a solid, see A381671.

			0.8287977192520120215005810038129635758617830308723...

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

Cf. A273637 (sphericity).
Cf. isoperimetric quotient of other Platonic solids: A019673 (cube), A073010 (octahedron), A374772 (dodecahedron), A381671 (tetrahedron).
Cf. A000796, A002194, A374883.

First[RealDigits[Pi*GoldenRatio^4/(15*Sqrt[3]), 10, 100]]

Equals Pi*phi^4/(15*sqrt(3)) = A000796*A374883/(15*A002194).

A384683 Decimal expansion of Sum_{i >= 1} 1/(3i-1) - 1/(3i).

Original entry on oeis.org

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

Offset: 0

Jason Bard, Jun 06 2025

Generalization of infinite sum generating A002162 (natural logarithm of 2). That sum is Sum_{i >= 1} 1/(k*i-1) - 1/(k*i), where k = 2. Here, we set k = 3.

			0.24700625029501853726527624218757023027640090422925...

Jason Bard, Table of n, a(n) for n = 0..9999
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 291.
Index entries for transcendental numbers.

Cf. A002162, A007494, A152743, A156057, A294514, A381671.

RealDigits[1/2 * Log[3] - (Sqrt[3]/18) * Pi, 10, 1000][[1]]

log(3)/2 - Pi/(6*sqrt(3)) \\ Amiram Eldar, Jun 07 2025

Equals (1/2) * log(3) - sqrt(3) * Pi / 18.
Equals Sum_{i>=1} 1/A152743(i).
Equals A294514/3. - Hugo Pfoertner, Jun 07 2025

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A093766 Decimal expansion of Pi/(2*sqrt(3)).

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A374772 Decimal expansion of the upper bound of the density of sphere packing in the Euclidean 3-space resulting from the dodecahedral conjecture.

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A306771 Numbers m such that m = i + j = i * k and phi(m) = phi(i) + phi(j) = phi(i) * phi(k) for some i, j, k, where phi is the Euler totient function A000010.

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A381672 Decimal expansion of the isoperimetric quotient of a regular icosahedron.

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A384683 Decimal expansion of Sum_{i >= 1} 1/(3*i-1) - 1/(3*i).

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A384683 Decimal expansion of Sum_{i >= 1} 1/(3i-1) - 1/(3i).