cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A080052 Value of n such that for any value of n, Pi^n is closer to its nearest integer than any value of Pi^k for 1 <= k < n.

Original entry on oeis.org

1, 2, 3, 58, 81, 157, 1030, 5269, 12128, 65875, 114791, 118885, 151710
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 22 2003

Keywords

Comments

Robert G. Wilson v used Mathematica with a changing number of digits to accommodate 24 digits to the right of the decimal point.
At 12128 the difference from an integer is 0.000016103224605297330719...
The sequence of rounded reciprocals of the distances, b(n) = round(1/(0.5-frac(Pi^a(n)-.5))) = round(1/abs(round(Pi^a(n))-Pi^a(n))), starts { 7, 8, 159, 190, 270, 2665, 10811, 26577, 62099, 70718, ... }. - M. F. Hasler, Apr 06 2008

Examples

			First term is 1 because this is just Pi = 3.14159....
Second term is 2 because Pi^2 = 9.869604... which is 0.13039... away from its nearest integer.
Pi^3 = 31.00627..., hence third term is 3.
Pi^58 is 0.00527... away from its nearest integer.

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 58, p. 21, Ellipses, Paris 2008.

Crossrefs

Cf. A079490, A137994, A137995.

Programs

  • Maple
    b := array(1..2000): Digits := 8000: c := 1: pos := 0: for n from 1 to 2000 do: exval := evalf(Pi^n): if (abs(exval-round(exval))
  • Mathematica
    a = 1; Do[d = Abs[ Round[Pi^n] - N[Pi^n, Ceiling[ Log[10, Pi^n] + 24]]]; If[d < a, Print[n]; a = d], {n, 1, 25000}]
$MaxExtraPrecision = 10^9; a = 1; Do[d = Abs[ Round[Pi^n] - N[Pi^n, Ceiling[ Log[10, Pi^n] + 24]]]; If[d < a, Print[n]; a = d], {n, 1, 10^5}] (* Ryan Propper, Nov 13 2005 *)
  • PARI
    f=0; for( i=1,99999, abs(frac(Pi^i)-.5)>f | next; f=abs(frac(Pi^i)-.5); print1(i",")) \\ M. F. Hasler, Apr 06 2008

Extensions

More terms from Carlos Alves and Robert G. Wilson v, Jan 23 2003
One more term from Ryan Propper, Nov 13 2005
a(11)-a(13) from Jeremy Elson, Nov 13 2011

A080279 Numbers n such that 1/G^n is closer to its nearest integer than any value of 1/G^k for 1 <= k < n, where G is Catalan's constant.

Original entry on oeis.org

1, 8, 52, 299, 437, 527, 2189, 64925
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 13 2003

Keywords

Comments

At n=2189 the discrepancy is 0.00000715379617...

Examples

			First term is 1 because this is just 1/G=1.0917440637... Second term is 8 because 1/G^8=2.01821167... which is 0.0182... away from its nearest integer. 1/G^52 is 0.0027 away from 96.

Crossrefs

Cf. A079490, A080052, A080053.

Programs

  • Maple
    a := []: s := 1: n := 1: do: g := 1/Catalan^n: d := round( 30+evalf( ilog10( g ) ) ): b := evalf(g,d): c := round(b): f := evalf(abs(c-b),d): if f

Extensions

More terms from Michel ten Voorde Jun 20 2003

A080053 Exp(n) is further from an integer than any previous exp(k) for 1 <= k < n.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 16, 21, 85, 115, 118, 136, 169, 177, 346, 1272, 2624, 8823, 12504, 13863, 36507, 51099, 63179, 111473, 143325, 153014, 255220, 476129
Offset: 1

Views

Author

Robert G. Wilson v, Jan 20 2003

Keywords

Examples

			Exp(4) = 54.59815... is closer to an integer than exp(2) = 7.389056... but not exp(3) = 20.085536...
The discrepancy at 111473 is 0.499991807891326554242475...

Crossrefs

Cf. A079490.

Programs

  • Mathematica
    a = 0; Do[ d = Abs[ Round[E^n] - N[E^n, Ceiling[ Log[10, E^n] + 10]]]; If[d > a, Print[n]; a = d], {n, 1, 1000000}]

Extensions

a(25)-a(28) from Martin Schlueter, Feb 08 2014

A080280 Numbers n such that Pi^(n*e)-e^n is closer to its nearest integer than any value of Pi^(k*e)-e^k for 1 <= k < n.

Original entry on oeis.org

1, 2, 5, 19, 212, 233, 299, 519, 1707, 3587, 8841, 8982, 12894
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 13 2003

Keywords

Comments

At 1707 the discrepancy is 0.0000981788...

Examples

			First term is 1 because this is just Pi^e-e=19.740875... Second term is 2 because Pi^(2*e)-e^2=497.0247 ...

Crossrefs

Cf. A079490, A080052, A080279.

Extensions

More terms from Michel ten Voorde Jun 20 2003

A080283 Numbers n such that (log(n)/Pi)^2 is closer to its nearest integer than any value of (log(k)/Pi)^2 for 1 <= k < n.

Original entry on oeis.org

2, 22, 23, 85, 2198, 83048, 422151, 2508952, 6635624, 199148648, 24591257752, 39660184000219160, 262537412640768744, 14468071444687145223825854225, 75579535015741588088534584527, 101634035376709910404057715634
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 13 2003

Keywords

Comments

(log(2198)/Pi)^2 = 6.0000064777...

Crossrefs

Cf. A079490, A080052, A080279-A080282.

Programs

  • PARI
    print1("2,22,");k=vector(2);d=1;for(n=1,500,k[1]=floor(exp(sqrt(n)*Pi)); k[2]=k[1]+1;for(i=1,2,s=(log(k[i])/Pi)^2;s=abs(s-round(s)); if(sRobert Gerbicz, Aug 24 2006

Extensions

More terms from Michel ten Voorde Jun 20 2003
Further terms from Robert Gerbicz, Aug 24 2006

A080284 Numbers n such that (Pi/e)^n is closer to its nearest integer than any value of (Pi/e)^k for 1 <= k < n.

Original entry on oeis.org

1, 5, 44, 49, 93, 94, 204, 283, 338, 547, 919, 1512, 1904, 22563, 52490, 98174
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 13 2003

Keywords

Comments

At n=1904, the discrepancy from an integer is 0.00003532...

Crossrefs

Cf. A079490, A080052, A080279-A080283.

Extensions

More terms from Michel ten Voorde Jun 20 2003

A080281 Numbers k such that Pi^k - 1/phi is closer to its nearest integer than any value of Pi^j - 1/phi for 1 <= j < k.

Original entry on oeis.org

1, 2, 4, 8, 17, 19, 23, 35, 221, 424, 3846, 16708, 19142, 19937, 55188, 87368
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 13 2003

Keywords

Comments

phi is the Golden ratio (1 + sqrt(5))/2.
At k = 3846 the discrepancy is 0.0000887984081945...
From Ryan Propper, Jul 27 2005: (Start)
At n = 16708 the discrepancy from an integer is 0.00006159...
At n = 19142 the discrepancy from an integer is 0.00003501...
At n = 19937 the discrepancy from an integer is 0.00001498...
At n = 55188 the discrepancy from an integer is 0.00001048...
At n = 87368 the discrepancy from an integer is 0.00000693...
(End)
As 1/phi = phi - 1, the sequence is equivalent to "Numbers k such that Pi^k - phi is closer to its nearest integer than any value of Pi^j - phi for 1 <= j < k." - David A. Corneth, Nov 19 2018

Examples

			The first term is 1 because this is just Pi - 1/phi  = 2.52355...
The second term is 2 because Pi^2 - 1/phi = 9.25157...
The next term is 4 because Pi^4 - 1/phi is closer to an integer than Pi^3 - 1/phi.

Crossrefs

Cf. A079490, A080052, A080279, A080280.

Programs

  • Mathematica
    $MaxExtraPrecision = 10^6; p = 2/(1+Sqrt[5]); b = 1; Do[a = Abs[N[Round[Pi^n - p] - (Pi^n - p), 30]]; If[a < b, Print[n]; b = a], {n, 1, 10^5}] (* Ryan Propper, Jul 27 2005 *)
  • PARI
    upto(n) = my(c = 2, phi = (1 + sqrt(5)) / 2, res = List, r = 2); Pik = 1; for(i = 1, n, Pik *= Pi; c = frac(Pik - phi); c = min(c, 1-c); if(c < r, listput(res, i); r = c)); res \\ David A. Corneth, Nov 19 2018

Extensions

a(12)-a(16) from Ryan Propper, Jul 27 2005

A080282 Numbers n such that log(n) + log_10(n) is closer to its nearest integer than any value of log(k) + log_10(k) for 1 < = k < n.

Original entry on oeis.org

2, 132, 264, 531, 8636, 69934, 140437, 282017, 566329, 1137266, 2283785, 4586151, 37138783, 74579724, 149766223, 603948755, 2435489735, 9821380043, 19722666141, 39605794500, 159714719422, 320728866517, 320728866518
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 13 2003

Keywords

Comments

log(320728866518) + log_10(320728866518) = 38.00000000000122913669...

Examples

			Interestingly, values of n that satisfy the criterion seem to start appearing in pairs if this sequence is extended further.
For example, the values 320728866517 and 320728866518 and later on the values 11169523543872502 and 11169523543872503. Many more of these n, n+1 pairs crop up in the sequence.

Crossrefs

Cf. A079490, A080052, A080279, A080280, A080281.

Extensions

More terms from Michel ten Voorde Jun 20 2003
Still more terms from Mark Hudson, Aug 26 2004

A080285 Numbers n such that [(Pi+e)/(Pi-e)]^n is closer to its nearest integer than any value of [(Pi+e)/(Pi-e)]^k for 1 <= k < n.

Original entry on oeis.org

1, 4, 10, 12, 20, 263, 964, 1533, 26974
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 13 2003

Keywords

Comments

At n=1533, the discrepancy from an integer is 0.000043205...
At n=26974, the discrepancy from an integer is 0.00002313... - Ryan Propper, Jul 26 2005

Crossrefs

Cf. A079490, A080052, A080279-A080284.

Programs

  • Mathematica
    $MaxExtraPrecision = 10^6; x = (Pi + E)/(Pi - E); b = 1; Do[a = Abs[N[Round[x^n] - x^n, 30]]; If[a < b, Print[n]; b = a], {n, 1, 30000}] (* Ryan Propper, Jul 26 2005 *)

Extensions

One more term from Ryan Propper, Jul 26 2005

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

Views

Author

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

Keywords

Comments

"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.

Examples

			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.

Crossrefs

Cf. A080052, A080053, A079490, A002160.

Programs

  • Maple
    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);
  • PARI
    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

Extensions

More terms from Robert Gerbicz, Aug 22 2006
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.