A066643 -id:A066643 - cp's OEIS frontend

A135039 Ceiling(Pi*n^2).

0, 4, 13, 29, 51, 79, 114, 154, 202, 255, 315, 381, 453, 531, 616, 707, 805, 908, 1018, 1135, 1257, 1386, 1521, 1662, 1810, 1964, 2124, 2291, 2464, 2643, 2828, 3020, 3217, 3422, 3632, 3849, 4072, 4301, 4537, 4779, 5027, 5282, 5542, 5809, 6083, 6362, 6648

Offset: 0

Mohammad K. Azarian, Feb 29 2008

Old name was "a(n)=ceiling[area of a circle of radius n]".

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

Cf. A066643.

Ceiling[Pi Range[50]^2] (* Harvey P. Dale, Oct 10 2011 *)

New name (using explicit formula), Joerg Arndt, Feb 19 2013

A004082 Numbers k such that sin(k-1) <= 0 and sin(k) > 0.

Original entry on oeis.org

1, 7, 13, 19, 26, 32, 38, 44, 51, 57, 63, 70, 76, 82, 88, 95, 101, 107, 114, 120, 126, 132, 139, 145, 151, 158, 164, 170, 176, 183, 189, 195, 202, 208, 214, 220, 227, 233, 239, 246, 252, 258, 264, 271, 277, 283, 290, 296

Offset: 1

Clark Kimberling

nonn

Apart from the first term this is also the sequence ceiling(circumference of a circle of radius n) = ceiling(2*Pi*n), n >= 1. - Mohammad K. Azarian, Feb 29 2008, Aug 01 2009

Bisection of A004084. - Michel Marcus, Mar 21 2013

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A004084, A066643.
For floor(2*Pi*n) see A038130.
See A277690 for another version.

Join[{1},Transpose[SequencePosition[Table[If[Sin[n]<=0,1,0],{n,300}],{1,0}]][[2]]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Apr 12 2016 *)

lista(m) = {for (i=1, m, if ((sin(i-1)<=0) && (sin(i) > 0), print1(i, ", ")););} \\ Michel Marcus, Mar 21 2013

a(n) = A038130(n-1) + 1.

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

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

A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

Original entry on oeis.org

0, 2, 8, 24, 42, 74, 104, 138, 186, 240, 304, 362, 424, 512, 594, 690, 776, 880, 986, 1104, 1232, 1346, 1490, 1624, 1762, 1930, 2088, 2256, 2418, 2594, 2784, 2962, 3170, 3368, 3584, 3810, 4008, 4248, 4466, 4730, 4976, 5210, 5474, 5736, 6024, 6306, 6570, 6864, 7154

Offset: 0

Felix Huber, Aug 05 2025

nonn

a(n) > 99% of the circle area for n >= 50.
Conjecture: The maximum possible area of a polygon within the circle would be the same if only the vertices but not the center were fixed on grid points.
All terms are even.

			See linked illustration of the term a(4) = 42.

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Illustration of a(4) = 42
Eric Weisstein's World of Mathematics, Pick's Theorem

Cf. A000217, A066643, A123690, A125228, A288247, A322106, A322107, A357577, A386539.

A386538:=proc(n)
    local x,y,p,s;
    p:=4*n;
    s:={};
    for x to n do
        y:=floor(sqrt(n^2-x^2));
        p:=p+4*y;
        s:=s union {y}
    od;
    return p-2*nops(s)
end proc;
seq(A386538(n),n=0..48);

a[n_] := Module[{p=4n},s = {}; Do[ y = Floor[Sqrt[n^2 - x^2]];p = p + 4*y;s = Union[s, {y}],{x,n} ];p - 2*Length[s]];Array[a,49,0] (* James C. McMahon, Aug 19 2025 *)

a(n) = A386539(A000217(n)) = A386539(n,n) for n >= 1.

a(n) <= A066643(n).

A135607 Floor of the area of a circle in terms of its circumference n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 20, 22, 25, 28, 31, 35, 38, 42, 45, 49, 53, 58, 62, 66, 71, 76, 81, 86, 91, 97, 103, 108, 114, 121, 127, 133, 140, 147, 154, 161, 168, 175, 183, 191, 198, 206, 215, 223, 232, 240, 249, 258, 267, 277, 286, 296, 305, 315

Offset: 0

Cino Hilliard, Feb 27 2008

nonn

			For a circle of circumference 10, the floor of the area A = floor(100/4/Pi) = 7.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A038130, A066643, A135039.

Table[Floor[n^2/(4*Pi)], {n,0,25}] (* G. C. Greubel, Oct 21 2016 *)

g(n) = for(c=0,n,a=c^2/4/Pi;print1(floor(a)","))

a(n) = n^2\(4*Pi); \\ Michel Marcus, Oct 22 2016

Area of a circle of radius r is A = Pi*r^2. Circumference of a circle of radius r is n = 2*Pi*r. Then area in terms of the circumference n is A = n^2/(4*Pi).

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).

A227794 Primes of the form floor(Pi*k^2).

Original entry on oeis.org

3, 113, 907, 3019, 3631, 5281, 6361, 7853, 8171, 11689, 14957, 16741, 17203, 20611, 33329, 36643, 38707, 63347, 68813, 96211, 115811, 126923, 128189, 129461, 169093, 172021, 234139, 241051, 248063, 301907, 319691, 340049, 367453, 380459, 382649, 387047, 448883

Offset: 1

K. D. Bajpai, Sep 23 2013

nonn

			a(2)=113: Pi*6^2 = 113.09 and 113 is prime.
a(3)=907: Pi*17^2 = 907.92 and 907 is prime.

Georg Fischer, Table of n, a(n) for n = 1..1500 [first 162 terms from _K. D. Bajpai_]

Cf. A066643 (floor(Pi*n^2)), A067559 (n that produce primes).

select(isprime, {seq(floor(Pi*n^2),n=1..1000)}); [corrected by Georg Fischer, Sep 27 2024]

Select[Floor[Pi*Range[400]^2],PrimeQ] (* Harvey P. Dale, Dec 18 2016 *)

is(n)=my(r=sqrtint((n+1)\Pi)); Pi*r^2>n && isprime(n) \\ Charles R Greathouse IV, Sep 23 2013

A235361 Floor((n + Pi)^2).

Original entry on oeis.org

9, 17, 26, 37, 51, 66, 83, 102, 124, 147, 172, 199, 229, 260, 293, 329, 366, 405, 446, 490, 535, 582, 632, 683, 736, 791, 849, 908, 969, 1033, 1098, 1165, 1234, 1306, 1379, 1454, 1532, 1611, 1692, 1775, 1861, 1948, 2037, 2129, 2222, 2317, 2414, 2514, 2615

Offset: 0

Alex Ratushnyak, Jan 07 2014

			a(1) = floor((Pi + 1)^2) = floor(17.1527897...) = 17.

Cf. A037085, A066643, A022844, A061294, A001672.

Table[Floor[(n + Pi)^2], {n, 0, 49}] (* Alonso del Arte, Jan 07 2014 *)

a(n) = floor((n+Pi)^2); \\ Michel Marcus, Jan 07 2014

A332084 Triangle read by rows: T(n,k) is the smallest m >= 0 such that floor(Pi*n^m) == k (mod n), -1 if one does not exist, k = 0..n-1.

Original entry on oeis.org

0, 1, 0, 0, 2, 4, 1, 3, 2, 0, 1, 7, 3, 0, 8, 1, 9, 14, 0, 10, 2, 1, 7, 10, 0, 8, 6, 2, 3, 1, 8, 0, 9, 6, 14, 5, 10, 1, 2, 0, 3, 20, 18, 11, 5, 32, 1, 6, 0, 2, 4, 7, 13, 11, 5, 5, 1, 8, 0, 13, 4, 2, 6, 9, 24, 12, 5, 1, 22, 0, 3, 17, 14, 18, 2, 6, 20, 10, 5, 1, 10, 0, 6, 9, 17, 14, 23, 7, 2, 21, 3

Offset: 1

Davis Smith, Aug 22 2020

Pi is normal in base n >= 2 if and only if in every row N, such that N is a power of n, -1 does not appear. Pi is absolutely normal if and only if -1 never appears.
Conjecture: Pi is absolutely normal, meaning that -1 will never appear.
This triangle is an instance of the more general f(n,k,r), where f(n,k,r) is the smallest m >= 0 such that floor(r*n^m) == k (mod n) (-1 if one does not exist) and r is irrational. The same conditions for normalcy apply.

			The triangle T(n,k) starts:
n\k   0   1   2   3   4   5   6   7   8   9  10  11  12 ...
1:    0
2:    1   0
3:    0   2   4
4:    1   3   2   0
5:    1   7   3   0   8
6:    1   9  14   0  10   2
7:    1   7  10   0   8   6   2
8:    3   1   8   0   9   6  14   5
9:   10   1   2   0   3  20  18  11   5
10:  32   1   6   0   2   4   7  13  11   5
11:   5   1  22   0  13   4   2   6   9  24  12
12:   5   1  10   0   3  17  14  18   2   6  20  10
13:   5   1  10   0   6   9  17  14  23   7   2  21   3

Davis Smith, Rows n = 1..144 of triangle, flattened
David G. Anderson, The Pi-Search Page.
D. H. Bailey, J. M. Borwein, C. S. Calude, M. J. Dinneen, M. Dumitrescu, and A. Yee, An Empirical Approach to the Normality of π, Experimental Math., 21 (2012), 375-384.
C. Sevcik, Fractal analysis of Pi normality, arXiv:1608.00430 [math.GM], 2016.
P. Trueb, Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi, arXiv:1612.00489 [math.NT], 2016.
Wikipedia, Normal number.

Cf. A000796, A022844, A066643, A068425, A176341.

Positions of 0 through 9 in base 10: A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008.

A332084_row(n)={my(L=List(vector(n,z,-1)), m=-1); while(vecmin(Vec(L))==-1, my(Z=lift(Mod(floor(Pi*n^(m++)),n))+1); if(L[Z]<0,listput(L,m,Z))); Vec(L)}

T(n,3) = 0, n > 3.

cp's OEIS Frontend

A135039 Ceiling(Pi*n^2).

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A004082 Numbers k such that sin(k-1) <= 0 and sin(k) > 0.

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

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

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A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

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A135607 Floor of the area of a circle in terms of its circumference n.

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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).