A001672 -id:A001672

A153713

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..15.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

Cf. A001672.

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

EXTENSIONS

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

STATUS

approved

A134915

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Coincides with first 11 terms of A085839.

LINKS

Table of n, a(n) for n=0..28.

FORMULA

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

MATHEMATICA

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

CROSSREFS

Cf. A085839, A001672.

Essentially the same as A115239.

KEYWORD

nonn

AUTHOR

Rolf Pleisch, Jan 29 2008

STATUS

approved

A138324

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A001672.

KEYWORD

nonn

AUTHOR

Leroy Quet, Mar 14 2008

EXTENSIONS

More terms from Vim Wenders, Mar 28 2008

STATUS

approved

A001674

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

T. D. Noe, Table of n, a(n) for n = 0..500

OEIS index entries related to powers of irrational constants.

MATHEMATICA

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

PROG

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

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by M. F. Hasler, May 29 2018

STATUS

approved

A071973

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000720, A001672, A061274.

KEYWORD

hard,nonn

AUTHOR

Robert G. Wilson v, Jun 18 2002

EXTENSIONS

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

STATUS

approved

A061294

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..39.

EXAMPLE

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

MAPLE

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

MATHEMATICA

a[n_]:=Floor[n^Pi]; (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)

Array[Floor[#^Pi]&, 40] (* Harvey P. Dale, Aug 29 2012 *)

CROSSREFS

Cf. A001672.

KEYWORD

nonn,easy

AUTHOR

Amarnath Murthy, Apr 26 2001

EXTENSIONS

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

STATUS

approved

A137299

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

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The sequence was suggested by Leroy Quet.

LINKS

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.

EXAMPLE

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]

MATHEMATICA

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 *)

PROG

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

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

CROSSREFS

Cf. A001203, A001672, A138324.

KEYWORD

nonn,easy,tabl

AUTHOR

M. F. Hasler, Mar 14 2008

STATUS

approved

A121245

(Floor(n*Pi))^n.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

CROSSREFS

Cf. A001672.

KEYWORD

nonn,less

AUTHOR

Mohammad K. Azarian, Aug 22 2006

STATUS

approved

A137995

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

+10
1

7, 159, 270, 308, 745, 758, 1080, 1227, 7805

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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!

LINKS

Table of n, a(n) for n=1..9.

PROG

(PARI) default(realprecision, 10^4); f=1; for(i=1, 10^9, frac(Pi^i)<f || next; f=frac(Pi^i); print1(round(1/f), ", "))

CROSSREFS

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

KEYWORD

nonn,more

AUTHOR

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

STATUS

approved

A235361

Floor((n + Pi)^2).

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..48.

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Alex Ratushnyak, Jan 07 2014

STATUS

approved