A001672 -id:A001672 - cp's OEIS frontend

A153713 Greatest number m such that the fractional part of Pi^A137994(n) <= 1/m.

7, 159, 270, 307, 744, 757, 796, 1079, 1226, 7804, 13876, 62099, 70718, 86902, 154755

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(2)=159 since 1/160<fract(Pi^A137994(2))=fract(Pi^3)=0.0062766...<=1/159.

Cf. A081464, A153669, A153677, A153685, A153693, A153701, A137994, A154130, A153721.

Cf. A001672.

A137994 = {1, 3, 81, 264, 281, 472, 1147, 2081, 3207, 3592, 10479, 12128, 65875, 114791, 118885};
Table[fp = FractionalPart[Pi^A137994[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A137994]}] (* Robert Price, Mar 26 2019 *)

a(n) = floor(1/fract(Pi^A137994(n))), where fract(x) = x-floor(x).

a(14)-a(15) from Robert Price, Mar 26 2019

A134915 a(0) = 1, a(n) = floor(a(n - 1)*Pi).

Original entry on oeis.org

1, 3, 9, 28, 87, 273, 857, 2692, 8457, 26568, 83465, 262213, 823766, 2587937, 8130243, 25541911, 80242279, 252088554, 791959549, 2488014301, 7816327450, 24555716894, 77144059797, 242355211526, 761381352089, 2391950062303, 7514532743484, 23607600862089, 74165465437218

Offset: 0

Rolf Pleisch, Jan 29 2008

nonn

Coincides with first 11 terms of A085839.

Cf. A085839, A001672.

Essentially the same as A115239.

NestList[Floor[# \[Pi]]&, 1, 30]  (* Harvey P. Dale, Mar 28 2011 *)

a(n) = A115239(n), n > 0. [From R. J. Mathar, Oct 27 2008]

A138324 a(n) = the second term in the simple continued fraction of Pi^n.

Original entry on oeis.org

7, 1, 159, 2, 50, 2, 3, 1, 10, 21, 55, 5, 3, 5, 1, 1, 1, 14, 1, 12, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 11, 2, 1, 1, 3, 3, 1, 1, 1, 2, 1, 1, 1, 7, 1, 8, 1, 2, 33, 1, 1, 1, 1, 117, 1, 2, 1, 1, 1, 8, 1, 2, 1, 1, 1, 1, 1, 27, 1, 1, 5, 4, 1, 1, 1, 270, 1, 1, 1, 5, 3, 1, 25, 2, 10, 9, 1, 16, 1, 1, 1

Offset: 1

Leroy Quet, Mar 14 2008

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A001672.

Table[Floor[1/(# - Floor[#])] &[Pi^n], {n, 96}] (* Michael De Vlieger, Aug 29 2017 *)

\p 2000 a(n) = floor(1/(Pi^n - floor(Pi^n))); for(i=1,100,print1(a(i),",")) \\ Vim Wenders, Mar 28 2008

a(n) = floor(1/(pi^n - floor(pi^n))).

More terms from Vim Wenders, Mar 28 2008

A001674 a(n) = floor(sqrt( 2*Pi )^n).

Original entry on oeis.org

1, 2, 6, 15, 39, 98, 248, 621, 1558, 3906, 9792, 24546, 61528, 154230, 386597, 969056, 2429063, 6088760, 15262258, 38256809, 95895600, 240374623, 602529828, 1510318305, 3785806567, 9489609784, 23786924200, 59624976768, 149457652641, 374634777972

Offset: 0

N. J. A. Sloane

nonn

Cf. A001674 (ceiling sqrt(2 Pi)^n), A017910 (floor sqrt(2)^n), A000149 (floor e^n), A001672 (floor Pi^n), A062541 (floor (Pi*e)^n), A121831 (floor (Pi+e)^n), A032739 (floor (Pi/e)^n), A014217 (floor ((1+sqrt(5))/2)^n).

Table[Floor[Sqrt[2*Pi]^n], {n, 0, 50}] (* T. D. Noe, Aug 09 2012 *)

a(n)=(2*Pi)^(n/2)\1 \\ M. F. Hasler, May 29 2018

Edited by M. F. Hasler, May 29 2018

A071973 Number of primes less than or equal to Pi^n.

Original entry on oeis.org

0, 2, 4, 11, 25, 62, 162, 433, 1175, 3229, 9042, 25549, 73050, 210356, 610041, 1779830, 5218745, 15372304, 45455747, 134882577, 401480918, 1198344171, 3585783711, 10754085805, 32319203663, 97312548674, 293515297707, 886720888966, 2682778745396, 8127887397064

Offset: 0

Robert G. Wilson v, Jun 18 2002

David Baugh, Table of n, a(n) for n = 0..52 (terms n = 27..52 found using Kim Walisch's primecount program)
Index entries for sequences related to numbers of primes in various ranges

Cf. A000720, A001672, A061274.

Mathematica
```
Do[ Print[ PrimePi[Pi^n]], {n, 0, 28}]
```

a(n) = primepi(Pi^n); \\ Michel Marcus, Oct 05 2020

a(n) = A000720(A001672(n)). - Michel Marcus, Oct 05 2020

a(27)-a(29) from David Baugh, Oct 05 2020

A061294 a(n) = floor( n^Pi ).

Original entry on oeis.org

1, 8, 31, 77, 156, 278, 451, 687, 995, 1385, 1869, 2456, 3159, 3987, 4952, 6065, 7337, 8781, 10406, 12226, 14251, 16494, 18966, 21680, 24646, 27878, 31387, 35186, 39287, 43703, 48445, 53526, 58959, 64756, 70930, 77494, 84459, 91840, 99649

Offset: 1

Amarnath Murthy, Apr 26 2001

			a(5) = floor(5^Pi) = floor(156.992545308865907578459198832649...) = 156.

Cf. A001672.

for n from 1 to 50 do printf("%d,",floor(n^Pi)) od;

a[n_]:=Floor[n^Pi]; (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)
Array[Floor[#^Pi]&,40] (* Harvey P. Dale, Aug 29 2012 *)

More terms from Winston C. Yang (winston(AT)cs.wisc.edu), May 19 2001

A137299 Square matrix read by antidiagonals: T(m,n) = m-th term in the continued fraction expansion of Pi^n.

Original entry on oeis.org

3, 9, 7, 31, 1, 15, 97, 159, 6, 1, 306, 2, 3, 1, 292, 961, 50, 2, 7, 2, 1, 3020, 2, 1, 3, 1, 47, 1, 9488, 3, 1, 4, 1, 13, 1, 1, 29809, 1, 2, 1, 60, 16539, 2, 8, 2, 93648, 10, 1, 2, 3, 1, 1, 1, 1, 1, 294204, 21, 14, 7, 3, 9, 4, 6, 3, 1, 3, 924269, 55, 15, 1, 1, 2, 1, 23, 7, 1, 2, 1

Offset: 1

M. F. Hasler, Mar 14 2008

The sequence was suggested by Leroy Quet.

			The matrix limited to order 10 is given by matrix(10,10,m,n,contfrac(Pi^n)[m]):
[   3   9   31    97   306   961  3020  9488 29809 93648]
[   7   1  159     2    50     2     3     1    10    21]
[  15   6    3     2     1     1     2     1    14    15]
[   1   1    7     3     4     1     2     7     1     1]
[ 292   2    1     1    60     3     3     1     9     4]
[   1  47   13 16539     1     9     2     1     3     2]
[   1   1    2     1     4     1    10     3     1     1]
[   1   8    1     6    23     5     4     1     5     3]
[   2   1    3     7     1     1     1     1     8     2]
[   1   1    1     6     2     3     1     1    16     1]

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)
J. S. Markovitch, Coincidence, data compression and Mach's concept of "economy of thought", APRI-PH-2004-12b, June 3 2004.

Cf. A001203, A001672, A138324.

A137299list[dmax_]:=With[{a=Array[ContinuedFraction[Pi^(dmax+1-#),#]&,dmax]},Array[Diagonal[a,#]&,dmax,1-dmax]];A137299list[10] (* Generates 10 antidiagonals *) (* Paolo Xausa, Nov 14 2023 *)

concat(vector(20,i,vector(i,j,contfrac(Pi^(i-j+1))[j])))

PARI
```
T(m,n)=contfrac(Pi^n)[m]
```

A121245 (Floor(n*Pi))^n.

Original entry on oeis.org

3, 36, 729, 20736, 759375, 34012224, 1801088541, 152587890625, 10578455953408, 819628286980801, 70188843638032384, 6582952005840035281, 671088640000000000000, 73885357344138503765449

Offset: 1

Mohammad K. Azarian, Aug 22 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A001672.

Table[Floor[(Pi n)]^n, {n, 20}] (* Vincenzo Librandi, Feb 22 2013 *)

A137995 Nearest integer to 1/frac(Pi^A137994(n)), where frac(x) = x - floor(x).

Original entry on oeis.org

7, 159, 270, 308, 745, 758, 796, 1080, 1227, 7805, 13876, 62099, 70718, 86902, 154756

Offset: 1

M. F. Hasler, inspired by Leroy Quet, Apr 05 2008

Sequence A137994 could be defined as "least positive integer such that this one (without rounding) is increasing".

The term a(1)=7 is not surprising (3 + 1/7 = 3.14...) but it comes as a funny surprise that the next term, a(2)=159, matches the next 3 digits of Pi and a(3) just differs by 5 from the next 3 digits!

Cf. A137994, A001203, A001672, A137299, A138324.

default(realprecision,10^4); f=1; for(i=1,10^9, frac(Pi^i)

a(7) inserted and a(11)-a(15) added by Amiram Eldar, Jun 28 2025

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

cp's OEIS Frontend

A153713 Greatest number m such that the fractional part of Pi^A137994(n) <= 1/m.

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A134915 a(0) = 1, a(n) = floor(a(n - 1)*Pi).

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A138324 a(n) = the second term in the simple continued fraction of Pi^n.

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A001674 a(n) = floor(sqrt( 2*Pi )^n).

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A071973 Number of primes less than or equal to Pi^n.

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A061294 a(n) = floor( n^Pi ).

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A137299 Square matrix read by antidiagonals: T(m,n) = m-th term in the continued fraction expansion of Pi^n.

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A121245 (Floor(n*Pi))^n.

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