A138324 -id:A138324 - cp's OEIS frontend

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.

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

A137299 Square matrix read by antidiagonals: T(m,n) = m-th term in the continued fraction expansion of Pi^n.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Mar 14 2008

The sequence was suggested by Leroy Quet.

			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]

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.

Cf. A001203, A001672, A138324.

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 *)

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

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

A057213 Second term of continued fraction for exp(n).

Original entry on oeis.org

1, 2, 11, 1, 2, 2, 1, 1, 11, 2, 7, 1, 2, 3, 2, 1, 1, 7, 1, 2, 2, 7, 4, 1, 2, 1, 1, 2, 23, 2, 2, 1, 15, 1, 1, 4, 3, 4, 1, 2, 4, 2, 2, 1, 28, 7, 3, 2, 10, 2, 6, 1, 1, 2, 1, 6, 1, 1, 3, 3, 9, 26, 3, 1, 2, 16, 2, 2, 1, 8, 1, 2, 1, 1, 69, 1, 1, 2, 2, 1, 5, 3, 1, 2, 2, 1, 1, 1, 1, 1, 68, 1, 1, 1, 1, 1, 2

Offset: 1

Leroy Quet, Sep 29 2000

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

Cf. A138324.

Table[ContinuedFraction[Exp[n],3][[2]],{n,100}] (* Harvey P. Dale, Nov 15 2014 *)

default(realprecision, 10000);
for(n=1, 10000, print(n, " ", contfrac(exp(n))[2]))
\\ Eric M. Schmidt, Mar 05 2014

floor(1/(exp(n)-floor(exp(n))))

A137995 Nearest integer to 1/frac(Pi^A137994(n)), where frac(x) = x - floor(x).

Original entry on oeis.org

7, 159, 270, 308, 745, 758, 796, 1080, 1227, 7805, 13876, 62099, 70718, 86902, 154756

Offset: 1

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

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!

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

default(realprecision,10^4); f=1; for(i=1,10^9, frac(Pi^i)

a(7) inserted and a(11)-a(15) added by Amiram Eldar, Jun 28 2025

cp's OEIS Frontend

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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A137299 Square matrix read by antidiagonals: T(m,n) = m-th term in the continued fraction expansion of Pi^n.

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A057213 Second term of continued fraction for exp(n).

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A137995 Nearest integer to 1/frac(Pi^A137994(n)), where frac(x) = x - floor(x).

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