A140758 -id:A140758 - cp's OEIS frontend

A022844 a(n) = floor(n*Pi).

0, 3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78, 81, 84, 87, 91, 94, 97, 100, 103, 106, 109, 113, 116, 119, 122, 125, 128, 131, 135, 138, 141, 144, 147, 150, 153, 157, 160, 163, 166, 169, 172, 175, 179, 182, 185, 188, 191, 194

Offset: 0

Clark Kimberling

nonn

Beatty sequence for Pi.
Differs from A127451 first at a(57). - L. Edson Jeffery, Dec 01 2013
These are the nonnegative integers m satisfying sin(m)*sin(m+1) <= 0. In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying sin(m*x)*sin((m+1)*x) <= 0, where x = Pi/r. Thus the numbers m satisfying sin(m*x)*sin((m+1)*x) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 21 2014
This can also be stated in terms of the tangent function. These are the nonnegative integers m such that tan(m/2)*tan(m/2+1/2) <= 0. In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying tan(m*x/2)*tan((m+1)*x/2) <= 0, where x = Pi/r. Thus the numbers m satisfying tan(m*x/2)*tan((m+1)*x/2) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 22 2014

			a(7)=21 because 7*Pi=21.9911... and a(8)=25 because 8*Pi=25.1327.... a(100000)=314159 because Pi=3.141592...

Ivan Panchenko, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A000796, A054386, A038130, A108591, A127451, A140758.

First differences give A063438.

R:=RieldField(10); [Floor(n*Pi(R)): n in [0..80]]; // G. C. Greubel, Sep 28 2018

a:=n->floor(n*Pi): seq(a(n),n=0..70); # Muniru A Asiru, Sep 28 2018

Floor[Pi Range[0,200]] (* Harvey P. Dale, Aug 27 2024 *)

vector(80, n, n--; floor(n*Pi)) \\ G. C. Greubel, Sep 28 2018

a(n)/n converges to Pi because |a(n)/n - Pi| = |a(n) - n*Pi|/n < 1/n. - Hieronymus Fischer, Jan 22 2006

Previous Mathematica program replaced by Harvey P. Dale, Aug 27 2024

A038130 Beatty sequence for 2*Pi.

Original entry on oeis.org

0, 6, 12, 18, 25, 31, 37, 43, 50, 56, 62, 69, 75, 81, 87, 94, 100, 106, 113, 119, 125, 131, 138, 144, 150, 157, 163, 169, 175, 182, 188, 194, 201, 207, 213, 219, 226, 232, 238, 245, 251, 257, 263, 270, 276, 282, 289, 295, 301, 307, 314, 320, 326, 333, 339, 345

Offset: 0

Felice Russo

nonn

a(n) = floor[circumference of a circle of radius n]. - Mohammad K. Azarian, Feb 29 2008

This sequence consists of the nonnegative integers k satisfying sin(k) <= 0 and sin(k+1) >= 0; thus this sequence and A246388 partition A022844 (the Beatty sequence for Pi). - Clark Kimberling, Aug 24 2014

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

Complement of A108586.
For ceiling (2*Pi*n) see A004082.
Cf. A022844, A140758, A108592, A246388.

Table[Floor[2 n*Pi], {n, 0, 100}] (* or *)
Select[Range[0, 628], Sin[#] <= 0 && Sin[# + 1] >= 0 &] (* Clark Kimberling, Aug 24 2014 *)

a(n) = floor(2*Pi*n).

a(n) = A004082(n+1) - 1. - John W. Nicholson, Mar 20 2025

More terms from Mohammad K. Azarian, Feb 29 2008

A108586 Floor(2nPi/(2*Pi-1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset: 1

Reinhard Zumkeller, Jun 11 2005

nonn

Beatty sequence for 2*Pi/(2*Pi-1); complement of A038130; not the same as A108120: a(37)=44 <> A108120(37)=43.

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

Cf. A054386, A140758, A108592.

With[{c=2Pi},Floor[(c*Range[80])/(c-1)]] (* Harvey P. Dale, Apr 21 2024 *)

A108589 a(n) = floor(n*Pi/(Pi-2)).

Original entry on oeis.org

2, 5, 8, 11, 13, 16, 19, 22, 24, 27, 30, 33, 35, 38, 41, 44, 46, 49, 52, 55, 57, 60, 63, 66, 68, 71, 74, 77, 79, 82, 85, 88, 90, 93, 96, 99, 101, 104, 107, 110, 112, 115, 118, 121, 123, 126, 129, 132, 134, 137, 140, 143, 145, 148, 151, 154, 156, 159, 162, 165, 167

Offset: 1

Reinhard Zumkeller, Jun 11 2005

Beatty sequence for Pi/(Pi-2); complement of A140758.

Vincenzo Librandi, Table of n, a(n) for n = 1..7000
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A054386, A108586, A108590, A140758.

R:= RealField(40); [Floor(n*Pi(R)/(Pi(R)-2)): n in [1..60]]; // G. C. Greubel, Oct 21 2023

A108589:=n->floor(n*Pi/(Pi-2)); seq(A108589(n), n=1..50); # Wesley Ivan Hurt, Apr 19 2014

With[{c=Pi/(Pi-2)},Floor[c*Range[70]]] (* Harvey P. Dale, Apr 19 2014 *)

[floor(n*pi/(pi-2)) for n in range(1,61)] # G. C. Greubel, Oct 21 2023

A108590 Self-inverse integer permutation induced by Beatty sequences for Pi/2 and Pi/(Pi-2).

Original entry on oeis.org

2, 1, 5, 8, 3, 11, 13, 4, 16, 19, 6, 22, 7, 24, 27, 9, 30, 33, 10, 35, 38, 12, 41, 14, 44, 46, 15, 49, 52, 17, 55, 57, 18, 60, 20, 63, 66, 21, 68, 71, 23, 74, 77, 25, 79, 26, 82, 85, 28, 88, 90, 29, 93, 96, 31, 99, 32, 101, 104, 34, 107, 110, 36, 112, 115, 37, 118, 39, 121

Offset: 1

Reinhard Zumkeller, Jun 11 2005

nonn

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to Beatty sequences

Cf. A108591, A108592.

a(A140758(n))=A108589(n) and a(A108589(n))=A140758(n).

A189515 a(n) = n + [ns] + [nt]; s=Pi/2, t=2/Pi.

Original entry on oeis.org

2, 6, 8, 12, 15, 18, 21, 25, 28, 31, 35, 37, 41, 43, 47, 51, 53, 57, 60, 63, 66, 70, 73, 76, 79, 82, 86, 88, 92, 96, 98, 102, 105, 108, 111, 114, 118, 121, 124, 127, 131, 133, 137, 141, 143, 147, 149, 153, 156, 159, 163, 166, 169, 172, 176, 178, 182, 185, 188, 192, 194, 198, 201, 204, 208, 211, 214, 217, 220, 223, 227, 230, 233, 237, 239, 243, 246, 249, 253, 255, 259, 262, 265

Offset: 1

Clark Kimberling, Apr 23 2011

nonn

This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
  a(n) = n + [ns/r] + [nt/r],
  b(n) = n + [nr/s] + [nt/s],
  c(n) = n + [nr/t] + [ns/t], where []=floor.
Taking r=1, s=Pi/2, t=2/Pi gives a=A189515, b=A189516, c=A189517.

Cf. A189516, A189517.

r=1; s=Pi/2; t=2/Pi;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (*A189515*)
Table[b[n], {n, 1, 120}]  (*A189516*)
Table[c[n], {n, 1, 120}]  (*A189517*)

a(n) = n + A140758(n) + floor(2*n/Pi). - R. J. Mathar, Sep 30 2011

A372610 Decimal expansion of Sum_{k >= 1} floor((Pi/2)*k)/2^k.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, May 07 2024

			2.669291338582677165354330708661417322834645669291338582677...

Paolo Xausa, Table of n, a(n) for n = 1..10000
J. M. Borwein and P. B. Borwein, Strange Series and High Precision Fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640.

Cf. A000796, A019669, A140758.

First[RealDigits[Sum[Floor[Pi*k/2]/2^k, {k, 400}], 10, 100]]

Approximately 339/127, correct to 67 digits: see Example 4.1 (a) in Borwein and Borwein (1992), p. 633.

A292987 Beatty sequence of the real root of x^5 - x^4 - x^2 - 1; complement of A292988.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 14, 15, 17, 18, 20, 21, 23, 25, 26, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 43, 45, 47, 48, 50, 51, 53, 54, 56, 58, 59, 61, 62, 64, 65, 67, 69, 70, 72, 73, 75, 76, 78, 80, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 98, 100, 102, 103, 105, 106, 108, 109, 111, 113, 114, 116, 117, 119, 120, 122, 124, 125, 127, 128, 130, 131, 133, 135, 136, 138, 139, 141, 142, 144, 146, 147, 149, 150, 152, 153, 155, 157, 158, 160, 161, 163, 164, 166, 168, 169, 171, 172, 174, 175, 177, 178

Offset: 1

Iain Fox, Dec 08 2017

nonn

First differs from A187342 at n = 37.

First differs from A140758 at n = 114.

			a(2) = floor(2 * 1.5701...) = floor(3.1402...) = 3.

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

Cf. A140758, A187342, A293506.

Complement: A292988.

r = N[Root[#^5 - #^4 - #^2 - 1 &, 1], 64]; Array[ Floor[r #] &, 70] (* Robert G. Wilson v, Dec 10 2017 *)

a(n) = floor(n*solve(x=1, 2, x^5 - x^4 - x^2 - 1))

a(n) = floor(n * r), where r = 1.57014731219605436291... (see A293506).

A132222 Beatty sequence 1+2floor(nPi/2), which contains infinitely many primes.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 25, 29, 31, 35, 37, 41, 43, 47, 51, 53, 57, 59, 63, 65, 69, 73, 75, 79, 81, 85, 87, 91, 95, 97, 101, 103, 107, 109, 113, 117, 119, 123, 125, 129, 131, 135, 139, 141, 145, 147, 151, 153, 157, 161, 163, 167, 169, 173, 175, 179, 183, 185, 189

Offset: 0

Jonathan Vos Post, Aug 14 2007

The primes in this entirely odd sequence begin 3, 7, 13, 19, 29, 31, 37, 41, 43, 47, 53, 59, 73, 79, 97, 101. By the theorems in Banks, there are an infinite number of primes in this sequence.

			a(0) = 1 because 1 + 2*floor(0*Pi) = 1 + 2*0 = 1 + 0 = 1.
a(1) = 3 because 1 + 2*floor(1*Pi/2) = 1 + 2*floor(1.5707963) = 1 + 2*1 = 3.
a(2) = 7 because 1 + 2*floor(2*Pi/2) = 1 + 2*floor(3.1415926) = 1 + 2*3 = 7.
a(3) = 9 because 1 + 2*floor(3*Pi/2) = 1 + 2*floor(4.7123889) = 1 + 2*4 = 9.
a(4) = 13 because 1 + 2*floor(4*Pi/2) = 1 + 2*floor(6.2831853) = 1 + 2*6 = 13.
a(5) = 15 because 1 + 2*floor(5*Pi/2) = 1 + 2*floor(7.8539816) = 1 + 2*7 = 15.
a(7) = 21 because 1 + 2*floor(7*Pi/2) = 1 + 2*floor(10.995574) = 1 + 2*10 = 21.

William D. Banks and Igor E. Shparlinski, Prime numbers with Beatty sequences, arXiv:0708.1015 [math.NT], 2007.

Cf. A019669, A130568, A140758.

Table[1 + 2*Floor[n*Pi/2], {n, 0, 60}] (* Stefan Steinerberger, Sep 02 2007 *)

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

a(n) = 1 + 2*A140758(n). - L. Edson Jeffery, Mar 16 2013

More terms from Stefan Steinerberger, Sep 02 2007

cp's OEIS Frontend

A022844 a(n) = floor(n*Pi).

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A038130 Beatty sequence for 2*Pi.

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A108586 Floor(2*n*Pi/(2*Pi-1)).

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A108589 a(n) = floor(n*Pi/(Pi-2)).

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A108590 Self-inverse integer permutation induced by Beatty sequences for Pi/2 and Pi/(Pi-2).

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A189515 a(n) = n + [ns] + [nt]; s=Pi/2, t=2/Pi.

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A372610 Decimal expansion of Sum_{k >= 1} floor((Pi/2)*k)/2^k.

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A292987 Beatty sequence of the real root of x^5 - x^4 - x^2 - 1; complement of A292988.

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A108586 Floor(2nPi/(2*Pi-1)).

A132222 Beatty sequence 1+2floor(nPi/2), which contains infinitely many primes.