A022852 -id:A022852 - cp's OEIS frontend

A233208 A measure of quality (the higher the better) for the approximation to e by rationals A022852(n)/n.

3, 2, 6, 7, 2, 3, 35, 3, 2, 5, 10, 2, 2, 17, 4, 2, 4, 14, 2, 2, 11, 5, 2, 4, 23, 3, 2, 8, 5, 2, 3, 66, 3, 2, 7, 7, 2, 3, 76, 3, 2, 5, 8, 2, 3, 24, 4, 2, 5, 11, 2, 2, 14, 4, 2, 4, 17, 2, 2, 10, 5, 2, 3, 33, 3, 2, 8, 6, 2, 3, 502, 3, 2, 6, 7, 2, 3, 38, 3, 2, 5, 9, 2, 2, 18, 4, 2, 4, 13, 2, 2, 12, 5, 2, 4, 22

Offset: 1

Franz Vrabec, Dec 06 2013

nonn

a(n) is the greatest natural number such that abs( n*e-A022852(n) ) < 1/a(n). Trivially a(n)>=2. a(n)=2 iff n is in A191104 (easy proof).

			a(7) = 35 because floor(1/abs(7*e-19)) = floor(1/0.0279727...) = floor(35.749...) = 35.

Cf. A022852. For records see A233209, A007677.

a(n)=floor(1/abs(n*exp(1)-round(n*exp(1)))) \\ Ralf Stephan, Dec 13 2013

a(n) = floor( 1 / abs( n*e-A022852(n) ) ).

More terms from Ralf Stephan, Dec 13 2013

A022843 Beatty sequence for e: a(n) = floor(n*e).

Original entry on oeis.org

0, 2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29, 32, 35, 38, 40, 43, 46, 48, 51, 54, 57, 59, 62, 65, 67, 70, 73, 76, 78, 81, 84, 86, 89, 92, 95, 97, 100, 103, 106, 108, 111, 114, 116, 119, 122, 125, 127, 130, 133, 135, 138, 141, 144, 146, 149, 152, 154, 157, 160

Offset: 0

Clark Kimberling

nonn

a(n) <= A022852(n) <= A121384(n). - Reinhard Zumkeller, Mar 17 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A054385, A108599.

Cf. A001113 (e), A022852, A121384.

a022843 n = a022843_list !! n
a022843_list = map (floor . (* e) . fromIntegral) [0..] where e = exp 1
-- Reinhard Zumkeller, Jul 06 2013

[Floor(n*Exp(1)): n in [0..60]]; // G. C. Greubel, Sep 28 2018

A022843 := proc(n)
    floor(n*exp(1)) ;
end proc: # R. J. Mathar, Jan 25 2015

Mathematica
```
Table[ Floor[n*E], {n, 1, 61}]
```

for (n=0, 100, print1(floor(n*exp(1)),", ")) \\ Indranil Ghosh, Mar 21 2017

import math
from mpmath import mp, e
mp.dps = 100
print([int(math.floor(n*e)) for n in range(51)]) # Indranil Ghosh, Mar 21 2017

a(n)/n converges to e because |a(n)/n-e|=|a(n)-n*e|/n < 1/n. - Hieronymus Fischer, Jan 22 2006

A121384 a(n) = ceiling(n*e).

Original entry on oeis.org

0, 3, 6, 9, 11, 14, 17, 20, 22, 25, 28, 30, 33, 36, 39, 41, 44, 47, 49, 52, 55, 58, 60, 63, 66, 68, 71, 74, 77, 79, 82, 85, 87, 90, 93, 96, 98, 101, 104, 107, 109, 112, 115, 117, 120, 123, 126, 128, 131, 134, 136, 139, 142, 145, 147, 150, 153, 155, 158, 161, 164, 166

Offset: 0

Mohammad K. Azarian, Sep 06 2006

Because the difference between e=A001113 and the constant 1/(1-theta), theta = A102525, defined in A054414 is only 0.00877, the difference |a(n)-A054414(n)| increases approximately as 0.00877*n. - R. J. Mathar, Apr 14 2008

A022843(n) <= A022852(n) <= a(n). - Reinhard Zumkeller, Mar 17 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A022843, A022852.

a121384 = ceiling . (* exp 1) . fromIntegral
-- Reinhard Zumkeller, Mar 17 2015

Table[Ceiling[n * E], {n, 0, 80}] (* Vincenzo Librandi, Feb 22 2013 *)

A319795 a(n) = n^(n+1)/(n-1)^n for n>1, rounded to nearest integer.

Original entry on oeis.org

8, 10, 13, 15, 18, 21, 23, 26, 29, 31, 34, 37, 40, 42, 45, 48, 50, 53, 56, 59, 61, 64, 67, 69, 72, 75, 78, 80, 83, 86, 88, 91, 94, 97, 99, 102, 105, 107, 110, 113, 116, 118, 121, 124, 126, 129, 132, 135, 137, 140, 143, 145, 148, 151, 154, 156, 159, 162, 164

Offset: 2

Jim Singh, Sep 28 2018

Cf. A084363, A022852, A121384.

seq(round(n^(n+1)/(n-1)^n),n=1..100); # Muniru A Asiru, Sep 28 2018

for(n=2,30,print1(round(n^(n+1)/(n-1)^n),","))

cp's OEIS Frontend

A233208 A measure of quality (the higher the better) for the approximation to e by rationals A022852(n)/n.

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A022843 Beatty sequence for e: a(n) = floor(n*e).

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A121384 a(n) = ceiling(n*e).

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A319795 a(n) = n^(n+1)/(n-1)^n for n>1, rounded to nearest integer.

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Maple

PARI