A002160 -id:A002160 - cp's OEIS frontend

A001672 a(n) = floor(Pi^n).

1, 3, 9, 31, 97, 306, 961, 3020, 9488, 29809, 93648, 294204, 924269, 2903677, 9122171, 28658145, 90032220, 282844563, 888582403, 2791563949, 8769956796, 27551631842, 86556004191, 271923706893, 854273519913, 2683779414317, 8431341691876, 26487841119103

Offset: 0

N. J. A. Sloane

nonn

T. D. Noe, Table of n, a(n) for n = 0..300

See also A002160: closest integer to Pi^n.

Cf. A001673.

Table[Floor[Pi^n], {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)

A001672(n)=Pi^n\1 \\ An error message will say so if default(realprecision) must be increased. - M. F. Hasler, May 27 2018

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

A000227 Nearest integer to e^n.

Original entry on oeis.org

1, 3, 7, 20, 55, 148, 403, 1097, 2981, 8103, 22026, 59874, 162755, 442413, 1202604, 3269017, 8886111, 24154953, 65659969, 178482301, 485165195, 1318815734, 3584912846, 9744803446, 26489122130, 72004899337, 195729609429, 532048240602, 1446257064291, 3931334297144, 10686474581524

Offset: 0

N. J. A. Sloane

x = e^n is the location of the maximum of x^(1/x^(1/n)). One can define another sequence, c(n) as the value of the natural number k that maximizes k^(1/k^(1/n)). Empirically, despite the rounding, c(n) and a(n) match each other until at least n>24500 (see the link). - Stanislav Sykora, Jun 06 2014

Federal Works Agency, Work Projects Administration for the City of NY, Tables of the Exponential Function. National Bureau of Standards, Washington, DC, 1939.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 230.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
Stanislav Sykora, Comments on A000227

Cf. A000149 (floor e^n), A001671 (e^n rounded up), A002160 (nearest integer to Pi^n).

Digits := 40: [seq(round(exp(n)), n=0..30)];

Table[ Round[ N[E^n] ], {n, 0, 30} ]
Round[E^Range[0,30]] (* Harvey P. Dale, Jul 28 2025 *)

apply( A000227(n)=exp(n)\/1, [0..50]) \\ An error message will say so if default(realprecision) must be increased. - M. F. Hasler, May 27 2018

A001675 a(n) = round(sqrt( 2*Pi )^n).

Original entry on oeis.org

1, 3, 6, 16, 39, 99, 248, 622, 1559, 3907, 9793, 24546, 61529, 154230, 386598, 969056, 2429064, 6088760, 15262259, 38256810, 95895601, 240374624, 602529829, 1510318305, 3785806568, 9489609784, 23786924201, 59624976768, 149457652642, 374634777972

Offset: 0

N. J. A. Sloane

nonn

Cf. A001674 (floor sqrt(2 Pi)^n), A001698 (ceiling sqrt(2 Pi)^n).

Cf. A017911 (round sqrt(2)), A000227 (round e^n), A002160 (round Pi^n).

Table[Floor[Sqrt[2*Pi]^n + 1/2], {n, 0, 50}] (* T. D. Noe, Aug 09 2012 *)
Round[(Sqrt[2*Pi])^Range[0,30] ] (* Harvey P. Dale, Jun 05 2018 *)

apply( a(n)=(2*Pi)^(n/2)\/1, [0..40]) \\ M. F. Hasler, May 29 2018

A386725 a(n) is the nearest integer to n/log_10(Pi).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129

Offset: 0

Stefano Spezia, Jul 31 2025

Differs from A005843 for n > 43.

For n > 0, a(n) is the nonnegative integer such that abs(10^(n/a(n)) - Pi) is minimum.

			a(n) = 2*n corresponds to the Brahmagupta's approximation 10^(1/2) = sqrt(10) of Pi (cf. A010467).

Cf. A000796, A002160, A005843, A010467, A053511, A385570.

a[n_]:=Round[n/Log10[Pi]]; Array[a,65,0]

a(n) = round(n/log10(Pi)).

a(n) >= 2*n.

A080072 Values of n such that Pi^n is farther from its closest integer than any Pi^k for 1 <= k < n.

Original entry on oeis.org

1, 4, 8, 31, 61, 89, 200, 217, 257, 1366, 3642, 4926, 20265

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 24 2003

"Farthest from an integer" only really makes sense if we choose "nearest" or "farthest" integer. I chose nearest here. "Farthest from farthest" would just make "nearest to nearest" and would be sequence A080052. I think.

			E.g., Pi^1=3.14159265... Pi^2=9.869..., Pi^3=31.00627..., Pi^4=97.40909... so Pi^4 is farther from 97 (its closest integer) than Pi^3 is from 31, or Pi^2 is from 10.

Cf. A080052, A080053, A079490, A002160.

b := array(1..5000): Digits := 10000: c := 0: pos := 0: for n from 1 to 10000 do: exval := evalf(Pi^n): if (abs(exval-round(exval))>c) then c := (abs(exval-round(exval))): pos := pos+1: b[pos] := n: print(n):fi: od: seq(b[n],n=1..pos);

default(realprecision,20000);d=0.0;p=Pi;a=1;for(n=1,40000,a*=p; s=abs(a-round(a));if(s>d,d=s;print1(n,","))) \\ Robert Gerbicz, Aug 22 2006

More terms from Robert Gerbicz, Aug 22 2006

A092786 a(n) = round(n*Pi^n).

Original entry on oeis.org

3, 20, 93, 390, 1530, 5768, 21142, 75908, 268282, 936480, 3236244, 11091230, 37747805, 127710397, 429872190, 1440515533, 4808357581, 15994483255, 53039715042, 175399135922, 578584268700, 1904232092224, 6254245258553

Offset: 1

Jorge Coveiro, Apr 14 2004

nonn

Cf. A002160.

Digits:=30; seq( (round(x*Pi^x)), x=1..60);

Table[Round[n*Pi^n],{n,30}] (* Harvey P. Dale, Aug 01 2020 *)

A130860 Number of decimal places of Pi given by integer approximations of the form a^(1/n).

Original entry on oeis.org

0, 0, 2, 2, 4, 2, 3, 4, 5, 5, 6, 6, 6, 6, 9, 9, 9, 10, 10, 11, 11, 13, 12, 13, 13, 14, 14, 14

Offset: 1

Stephen McInerney (spmcinerney(AT)hotmail.com), Jul 22 2007

Approximations are rounded, not truncated; see the example for n=2. Note that this can produce anomalous results; e.g., 0.148 does not match 0.152 to 1-place accuracy, but does match it to 2-place accuracy. - Franklin T. Adams-Watters, Mar 29 2014

			a(8)=4 because 9489^(1/8) = 3.1416... is Pi accurate to 4 decimal places.
a(2)=0. 10^(1/2) = 3.16... rounded to one place is 3.2, while Pi to one place is 3.1.

Cf. A002160.

from math import pi, floor, ceil
def round(x):
    return math.floor(x + 0.5)
def decimal_places(x, y):
    dp = -1
    # Compare integer part, shift 1 dp
    while floor(x + 0.5) == floor(y + 0.5) and x and y:
        x = (x - floor(x)) * 10
        y = (y - floor(y)) * 10
        dp = dp + 1
    return dp
for n in range(1, 30):
    pi_to_the_n = pow(pi, n)
    pi_to_the_n_rnd = round(pi_to_the_n)
    pi_approx = pow(pi_to_the_n_rnd, 1.0 / n)
    dps = decimal_places(pi_approx, pi)
    print(dps)

a(n) is the number of decimal_places in (round(Pi^n))^1/n w.r.t. Pi.

Note that round(Pi^n) is the sequence A002160 (Nearest integer to Pi^n).

A305289 Powers of 2*Pi, rounded to the nearest integer.

Original entry on oeis.org

1, 6, 39, 248, 1559, 9793, 61529, 386598, 2429064, 15262259, 95895601, 602529829, 3785806568, 23786924201, 149457652642, 939070127125, 5900351625162, 37073002638414, 232936545470713, 1463583480006755, 9195966217409213, 57779959822545404, 363042194606444109, 2281061383037441740

Offset: 0

M. F. Hasler, May 29 2018

nonn

Index entries of sequences related to powers of irrational constants.

Cf. A001674 (floor(sqrt(2 Pi)^n)), A001675 (round sqrt(2 Pi)^n), A001698 (ceiling sqrt(2 Pi)^n), A017911 (round sqrt(2)^n), A000227 (round e^n), A002160 (round Pi^n).

Round[(2Pi)^Range[0,30]] (* Harvey P. Dale, Aug 12 2021 *)

PARI
```
apply( a(n)=(2*Pi)^n\/1, [0..40])
```

a(n) = round((2 Pi)^n) = A001675(2*n) >= A001674(2n).

cp's OEIS Frontend

A001672 a(n) = floor(Pi^n).

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A000227 Nearest integer to e^n.

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A001675 a(n) = round(sqrt( 2*Pi )^n).

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A386725 a(n) is the nearest integer to n/log_10(Pi).

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A080072 Values of n such that Pi^n is farther from its closest integer than any Pi^k for 1 <= k < n.

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A092786 a(n) = round(n*Pi^n).

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A130860 Number of decimal places of Pi given by integer approximations of the form a^(1/n).

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A305289 Powers of 2*Pi, rounded to the nearest integer.

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