A066645 -id:A066645 - cp's OEIS frontend

A164086 Beatty sequence for 4*Pi/3 = 4.1887902... .

4, 8, 12, 16, 20, 25, 29, 33, 37, 41, 46, 50, 54, 58, 62, 67, 71, 75, 79, 83, 87, 92, 96, 100, 104, 108, 113, 117, 121, 125, 129, 134, 138, 142, 146, 150, 154, 159, 163, 167, 171, 175, 180, 184, 188, 192, 196, 201, 205, 209, 213, 217, 222, 226, 230, 234, 238, 242

Offset: 1

Reinhard Zumkeller, Aug 11 2009

nonn

a(n) = A109238(n) for n <= 20;
complement of A164087;
a(n) = A164088(A164087(n)) and A164088(a(n)) = A164087(n);
a(A000578(n)) = A066645(n).

			a(3^3) = a(27) = 113 = (integer part of volume of sphere with radius=3) = A066645(3).

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A022844, A019699.

With[{c=4 \[Pi]/3},Floor[c #]&/@Range[70]]  (* Harvey P. Dale, Mar 19 2011 *)

fprec:100$
A164086:4*%pi/3$
ev(bfloat(A164086)); /* Martin Ettl, Nov 03 2012 */

a(n) = floor(4*n*Pi/3).

A135973 Ceiling(4/3Pin^3).

Original entry on oeis.org

0, 5, 34, 114, 269, 524, 905, 1437, 2145, 3054, 4189, 5576, 7239, 9203, 11495, 14138, 17158, 20580, 24430, 28731, 33511, 38793, 44603, 50966, 57906, 65450, 73623, 82448, 91953, 102161, 113098, 124789, 137259, 150533, 164637, 179595, 195433

Offset: 0

Mohammad K. Azarian, Mar 02 2008

Volume of a sphere of radius n, rounded up.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A066643, A066644, A066645, A135039.

Table[Ceiling[4/3*Pi * n^3], {n, 0, 60}] (* Vincenzo Librandi, Feb 19 2013 *)

a(n)=ceil(4/3*Pi*n^3) \\ Charles R Greathouse IV, Oct 10 2013

n=100 # change n for more values
[ceil(4/3*pi*r^3) for r in [0..n]] # Tom Edgar, Oct 10 2013

a(n) = A066645(n) + 1 for n > 0.

Definition replaced by Vincenzo Librandi, Feb 19 2013

0 prepended by T. D. Noe, Oct 10 2013

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).

A135971 Ceiling(4Pin^2).

Original entry on oeis.org

13, 51, 114, 202, 315, 453, 616, 805, 1018, 1257, 1521, 1810, 2124, 2464, 2828, 3217, 3632, 4072, 4537, 5027, 5542, 6083, 6648, 7239, 7854, 8495, 9161, 9853, 10569, 11310, 12077, 12868, 13685, 14527, 15394, 16287, 17204, 18146, 19114, 20107

Offset: 1

Mohammad K. Azarian, Mar 02 2008

The original definition was "a(n)=ceiling[surface area of a shpere of radius n]".

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A066643, A066644, A066645, A135039.

Table[Ceiling[4 Pi n^2], {n, 1, 50}] (* Vincenzo Librandi, Feb 19 2013 *)

Definition replaced by Bruno Berselli, Feb 19 2013

A297839 Numbers k > 0 that set a new record for the closeness of (4/3)Pik^3 to an integer.

Original entry on oeis.org

1, 3, 4, 14, 18, 23, 62, 95, 423, 5339, 12352, 108359, 129805, 5334194, 82007322, 90401717, 199671691, 434184265, 655956850, 44438886071

Offset: 1

Felix Fröhlich, Jan 07 2018

Integer radii such that the volume of the corresponding sphere is closer to an integer than for any smaller integer radius.

			k       | (4/3)*Pi*k^3                             | Deviation from integer
---------------------------------------------------------------------------
1       |                     4.188790204786390... | 0.188790204786390...
3       |                   113.097335529232556... | 0.097335529232556...
4       |                   268.082573106329023... | 0.082573106329023...
14      |                 11494.040321933856861... | 0.040321933856861...
18      |                 24429.024474314232222... | 0.024474314232222...
23      |                 50965.010421636019109... | 0.010421636019109...
62      |                998305.991926330990581... | 0.008073669009418...
95      |               3591364.001828731970435... | 0.001828731970435...
423     |             317036825.999590816501793... | 0.000409183498206...
5339    |          637482653747.999839504336479... | 0.000160495663520...
12352   |         7894060641354.000003942767448... | 0.000003942767448...
108359  |      5329464512150064.999997849950689... | 0.000000215004931...
129805  |      9161421693208264.000000035388795... | 0.000000035388795...
5334194 | 635762677398025211698.999999995151941... | 0.000000004848058...

David Consiglio, Jr., Python Program
Sean A. Irvine, Java program (github)

Cf. A066645, A135973, A254714, A297840.

closeness(n) = my(v=(4/3)*Pi*n^3); if(round(v) > v, return(round(v)-v), return(v-round(v)))
my(r=1, k=1, c=0); while(1, c=closeness(k); if(c < r, print1(k, ", "); r=c); k++)

a(15)-a(19) from Jon E. Schoenfield, Jan 07 2018

a(20) from David Consiglio, Jr., Mar 14 2023

A210519 a(n) = floor(volume of 4-sphere of radius n).

Original entry on oeis.org

0, 4, 78, 399, 1263, 3084, 6395, 11848, 20212, 32377, 49348, 72250, 102328, 140942, 189575, 249824, 323407, 412159, 518035, 643108, 789568, 959725, 1156007, 1380959, 1637248, 1927657, 2255086, 2622556, 3033205, 3490291

Offset: 0

Jon Perry, Jan 26 2013

nonn

The 4-sphere here refers to the geometric sphere, that is, 4 refers to the number of dimensions of the sphere.

The general formula for the volume of an n-sphere can be derived using (4)-(10) at the Mathworld link, and some explicit values for higher dimensional spheres are given at the Wikipedia link, section 2.4. Note that Wikipedia uses the topologic definition and calls this 4-sphere a 3-sphere.

Clay Math Institute, Poincaré Conjecture Press Release
Mathworld, Hypersphere
Mathworld, Poincaré's Conjecture
Wikipedia, N-sphere

Cf. A066643, A066645.

pi = Math.PI;
for (i = 0; i < 60; i++) document.write(Math.floor(pi*pi*i*i*i*i/2) + ", ");

Mathematica
```
Table[Floor[(Pi^2 n^4)/2], {n, 0, 29}]
```

a(n) = floor(1/2*Pi^2*n^4).

cp's OEIS Frontend

A164086 Beatty sequence for 4*Pi/3 = 4.1887902... .

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Maxima

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A135973 Ceiling(4/3*Pi*n^3).

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Sage

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

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Maple

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A135971 Ceiling(4*Pi*n^2).

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A297839 Numbers k > 0 that set a new record for the closeness of (4/3)*Pi*k^3 to an integer.

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PARI

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A210519 a(n) = floor(volume of 4-sphere of radius n).

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A135973 Ceiling(4/3Pin^3).

A135971 Ceiling(4Pin^2).

A297839 Numbers k > 0 that set a new record for the closeness of (4/3)Pik^3 to an integer.