A001672 -id:A001672 - cp's OEIS frontend

A115240 a(n) = A001672(n) - A115239(n).

0, 0, 3, 10, 33, 104, 328, 1031, 3241, 10183, 31991, 100503, 315740, 991928, 3116234, 9789941, 30756009, 96622854, 303549648, 953629346, 2995914948, 9411944394, 29568495367, 92892167824, 291829352014, 916808948392, 2880240257014

Offset: 1

Hieronymus Fischer, Jan 17 2006

nonn

			a(5) = A001672(5) - A115239(5) = 306 - 273 = 33.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001672, A115239.

a[1] = Floor[Pi]; a[n_] := a[n] = Floor[a[n - 1]*Pi]; Array[ Floor[Pi^# ] - a[ # ] &, 27] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jan 18 2006

A119604 Merged values of A014217 = {floor(((1+sqrt(5))/2)^n)}, A000149 = {floor(e^n)}, and A001672 = {floor(Pi^n)}, with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 29, 31, 46, 54, 76, 97, 122, 148, 199, 306, 321, 403, 521, 842, 961, 1096, 1364, 2206, 2980, 3020, 3571, 5777, 8103, 9349, 9488, 15126, 22026, 24476

Offset: 0

Bryan David Levenson, Jun 03 2006

			a(8)=6 because floor(((1+sqrt(5))/2)^4)=6, a(9)=7 because floor(e^2)=7 and a(10)=9 because floor(Pi^2)=9.

OEIS index entries related to powers of irrational constants.

Cf. A001672, A014217, A000149.

Edited by M. F. Hasler, May 29 2018

A137994 a(n) is the smallest integer > a(n-1) such that {Pi^a(n)} < {Pi^a(n-1)}, where {x} = x - floor(x), a(1)=1.

Original entry on oeis.org

1, 3, 81, 264, 281, 472, 1147, 2081, 3207, 3592, 10479, 12128, 65875, 114791, 118885

Offset: 1

Leroy Quet and M. F. Hasler, Mar 14 2008

The sequence was suggested by Leroy Quet on Pi day 2008, cf. A138324.
The next such number must be greater than 100000. - Hieronymus Fischer, Jan 06 2009
a(16) > 300,000. - Robert Price, Mar 25 2019

			a(3)=81, since {Pi^81}=0.0037011283.., but {Pi^k}>=0.0062766802... for 1<=k<=80; thus {Pi^81}<{Pi^k} for 1<=k<81. - _Hieronymus Fischer_, Jan 06 2009

Cf. A001203, A138324, A001672.

Cf. A081464, A153669, A153677, A153685, A153693, A153705, A153713, A154130, A153717. - Hieronymus Fischer, Jan 06 2009

$MaxExtraPrecision = 10000;
p = .999;
Select[Range[1, 5000],
If[FractionalPart[Pi^#] < p, p = FractionalPart[Pi^#]; True] &] (* Robert Price, Mar 12 2019 *)

default(realprecision,10^4); print1(a=1); for(i=1,100, f=frac(Pi^a); until( frac(Pi^a++)

a(11)-a(13) from Hieronymus Fischer, Jan 06 2009
Edited by R. J. Mathar, May 21 2010
a(14)-a(15) from Robert Price, Mar 12 2019

A002160 Nearest integer to Pi^n.

Original entry on oeis.org

1, 3, 10, 31, 97, 306, 961, 3020, 9489, 29809, 93648, 294204, 924269, 2903677, 9122171, 28658146, 90032221, 282844564, 888582403, 2791563950, 8769956796, 27551631843, 86556004192, 271923706894, 854273519914, 2683779414318, 8431341691876, 26487841119104, 83214007069230

Offset: 0

N. J. A. Sloane

			a(0) = 1 because Pi^0 = 1;
a(2) = 10 because Pi^2 = 9.8696...;
a(10) = 93648 because Pi^10 = 93648.047476...

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. 122.
J. T. Peters, Ten-Place Logarithm Table. Vols. 1 and 2, rev. ed. Ungar, NY, 1957, Vol. 1 (Appendix), p. 1.
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).

Cf. A000227 (e^n), A001672 (floor(Pi^n)), A001673 (ceiling(Pi^n)).

a := []: Digits := 1000: for n from 0 to 50 do: a := [op(a),round(Pi^n)]: od: seq(a[i+1],i=0..50);

Round[Pi^Range[0,40]] (* Harvey P. Dale, Jun 10 2024 *)

apply( A002160(n)=Pi^n\/1, [0..50]) \\ An error message will say so if default(realprecision) must be increased. - M. F. Hasler, May 27 2018

[round(pi^n) for n in range(0,29)] # Stefano Spezia, Jan 15 2025

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 29 2003

Edited by M. F. Hasler, May 27 2018

A153710 Numbers k such that the fractional part of Pi^k is less than 1/k.

Original entry on oeis.org

1, 3, 5, 9, 10, 11, 59, 81, 264, 281, 472, 3592, 10479, 12128, 65875, 118885

Offset: 1

Hieronymus Fischer, Jan 08 2009

Numbers k such that fract(Pi^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(17) > 300000. - Robert Price, Mar 25 2019

			a(4) = 9 since fract(Pi^9) = 0.0993... < 1/9, but fract(Pi^k) = 0.3891..., 0.2932..., 0.5310... for 6 <= k <= 8, which all are greater than 1/k.

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153714, A154130, A153718.

Cf. A001672.

Select[Range[1000], N[FractionalPart[Pi^#], 100] < (1/#) &]  (* G. C. Greubel, Aug 25 2016 *)

isok(k) = frac(Pi^k) < 1/k; \\ Michel Marcus, Feb 11 2014

a(16) from Robert Price, Mar 25 2019

A153712 Numbers k such that the fractional part of Pi^k is greater than 1-(1/k).

Original entry on oeis.org

1, 2, 15, 22, 58, 109, 157, 1030, 1071, 1274, 2008, 2322, 5269, 151710

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract(Pi^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(15) > 300000. - Robert Price, Mar 25 2019

			a(3) = 15, since fract(Pi^15) = 0.969... > 0.933... = 1 - (1/15), but fract(Pi^k) <= 1 - (1/k) for 3 <= k <= 14.

Cf. A153664, A153672, A153680, A153688, A153696, A153704, A153716, A154130, A153720.

Cf. A001672.

Select[Range[1000], N[FractionalPart[Pi^#], 100] > 1 - (1/#) &]  (* G. C. Greubel, Aug 25 2016 *)

a(14) from Robert Price, Mar 25 2019

A153711 Minimal exponents m such that the fractional part of Pi^m obtains a maximum (when starting with m=1).

Original entry on oeis.org

1, 2, 15, 22, 58, 157, 1030, 5269, 145048, 151710

Offset: 1

Hieronymus Fischer, Jan 06 2009

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of Pi^m is greater than the fractional part of Pi^k for all k, 1<=k

The next such number must be greater than 100000.

a(11) > 300000. - Robert Price, Mar 25 2019

			a(3)=15, since fract(Pi^15)= 0.9693879984..., but fract(Pi^k)<=0.8696... for 1<=k<=14; thus fract(Pi^15)>fract(Pi^k) for 1<=k<15 and 15 is the minimal exponent > 2 with this property.

Cf. A153663, A153671, A153679, A153687, A153695, A153707, A153715, A154130, A153719.

Cf. A001672.

$MaxExtraPrecision = 100000;
p = 0; Select[Range[1, 10000],
If[FractionalPart[Pi^#] > p, p = FractionalPart[Pi^#]; True] &] (* Robert Price, Mar 25 2019 *)

Recursion: a(1):=1, a(k):=min{ m>1 | fract(Pi^m) > fract(Pi^a(k-1))}, where fract(x) = x-floor(x).

a(9)-a(10) from Robert Price, Mar 25 2019

A153715 Greatest number m such that the fractional part of Pi^A153711(m) >= 1-(1/m).

Original entry on oeis.org

1, 7, 32, 53, 189, 2665, 10810, 26577, 128778, 483367

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(3) = 32, since 1-(1/33) = 0.9696... > fract(Pi^A153711(3)) = fract(Pi^15) = 0.96938... >= 0.96875 = 1-(1/32).

Cf. A153663, A153671, A153679, A153687, A153695, A091560, A153711, A154130, A153723.

Cf. A001672.

$MaxExtraPrecision = 100000;
A153711 = {1, 2, 15, 22, 58, 157, 1030, 5269, 145048, 151710};
Floor[1/(1-FractionalPart[Pi^A153711])] (* Robert Price, Apr 18 2019 *)

a(n) = floor(1/(1-fract(Pi^A153711(n)))), where fract(x) = x-floor(x).

a(9)-a(10) from Robert Price, Apr 18 2019

A153714 Greatest number m such that the fractional part of Pi^A153710(n) <= 1/m.

Original entry on oeis.org

7, 159, 50, 10, 21, 55, 117, 270, 307, 744, 757, 7804, 13876, 62099, 70718, 154755

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(2)=159 since 1/160<fract(Pi^A153710(2))=fract(Pi^3)=0.0062766...<=1/159.

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153710, A154130, A153722.

Cf. A001672.

A153710 = {1, 3, 5, 9, 10, 11, 59, 81, 264, 281, 472, 3592, 10479,
   12128, 65875, 118885};
Table[fp = FractionalPart[Pi^A153710[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153710]}] (* Robert Price, May 10 2019 *)

a(n) = floor(1/fract(Pi^A153710(n))), where fract(x) = x-floor(x).

a(16) from Robert Price, May 10 2019

A153716 Greatest number m such that the fractional part of Pi^A153712(n) >= 1-(1/m).

Original entry on oeis.org

1, 7, 32, 53, 189, 131, 2665, 10810, 2693, 1976, 3697, 4289, 26577, 483367

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(3) = 32, since 1-(1/33) = 0.9696... > fract(Pi^A153712(3)) = fract(Pi^15) = 0.96938... >= 0.96875 = 1-(1/32).

Cf. A153664, A153672, A153680, A153688, A153696, A153704, A153712, A154130, A153724.

Cf. A001672.

A153712 = {1, 2, 15, 22, 58, 109, 157, 1030, 1071, 1274, 2008, 2322,
   5269, 151710};
Table[Floor[1/(1 - FractionalPart[Pi^A153712[[n]]])], {n, 1,
Length[A153712]}] (* Robert Price, May 10 2019 *)

a(n) = floor(1/(1-fract(Pi^A153712(n)))), where fract(x) = x-floor(x).

a(14) from Robert Price, May 10 2019

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cp's OEIS Frontend

A115240 a(n) = A001672(n) - A115239(n).

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A119604 Merged values of A014217 = {floor(((1+sqrt(5))/2)^n)}, A000149 = {floor(e^n)}, and A001672 = {floor(Pi^n)}, with multiplicity.

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A137994 a(n) is the smallest integer > a(n-1) such that {Pi^a(n)} < {Pi^a(n-1)}, where {x} = x - floor(x), a(1)=1.

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A002160 Nearest integer to Pi^n.

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A153710 Numbers k such that the fractional part of Pi^k is less than 1/k.

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A153712 Numbers k such that the fractional part of Pi^k is greater than 1-(1/k).

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A153711 Minimal exponents m such that the fractional part of Pi^m obtains a maximum (when starting with m=1).

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A153715 Greatest number m such that the fractional part of Pi^A153711(m) >= 1-(1/m).

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