A080052 -id:A080052 - cp's OEIS frontend

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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.

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

Offset: 1

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

At n=2189 the discrepancy is 0.00000715379617...

			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.

Cf. A079490, A080052, A080053.

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

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

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Author

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

Keywords

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nonn

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

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

Keywords

nonn

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

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

Keywords

more
nonn

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

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

Keywords

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nonn

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

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

Keywords

nonn

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

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

Keywords

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nonn

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

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

Keywords

nonn
more

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

A080286 Numbers m such that (Pi+e)^m is closer to its nearest integer than any value of (Pi+e)^k for 1 <= k < m.

Original entry on oeis.org

1, 4, 14, 25, 83, 90, 92, 547, 966, 1027, 1472, 5055, 5283

Offset: 1

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

Keywords

more
nonn

Comments

At m=1472, the discrepancy from an integer is .00015493757803793815931...

Crossrefs

Cf. A079490, A080052, A080279-A080285.

Extensions

More terms from Michel ten Voorde Jun 20 2003

A267122 Numbers n such that 1.5^n is closer to an integer than 1.5^m for any 0 < m < n.

Original entry on oeis.org

1, 2, 4, 29, 46, 58, 95, 153, 157, 163, 455, 1060, 1256, 2677, 3328, 12429, 49304, 112896, 129638, 164000

Offset: 1

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Charles R Greathouse IV, Jan 11 2016

Keywords

nonn

Comments

Zudilin proves that the distance from 1.5^n to the nearest integer is at least 0.5803^n for large enough n; it seems that n > 4 suffices. (The "large enough" constant in the proof is effective but not explicit.)

Examples

1.5^29 = 127834.039... which is within 0.039... of an integer, yielding a(4) = 29. 1.5^46 = 125949925.968... which is within 0.031... of an integer, yielding a(5) = 46.

Links

Wadim Zudilin, A new lower bound for ∥(3/2)^k∥, J. Théor. Nombres Bordeaux 19:1 (2007), pp. 311-323.

Crossrefs

Cf. A079490, A080052.

Programs

PARI

f(x)=x=frac(x); if(x>1/2,1-x,x) t=r=1;for(n=1,1e6, tt=f(t*=3/2); if(tt

Showing 1-10 of 10 results.

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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.

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Examples

Crossrefs

Programs

Maple

Extensions

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.

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Extensions

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.

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PARI

Extensions

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.

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Extensions

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.

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Mathematica

PARI

Extensions

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.

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Extensions

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.

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Programs

Mathematica

Extensions

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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Examples

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Programs

Maple

PARI

Extensions

A080286 Numbers m such that (Pi+e)^m is closer to its nearest integer than any value of (Pi+e)^k for 1 <= k < m.

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Crossrefs

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