A022843 -id:A022843 - cp's OEIS frontend

A054385 Beatty sequence for e/(e-1); complement of A022843.

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 60, 61, 63, 64, 66, 68, 69, 71, 72, 74, 75, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 98, 99, 101, 102, 104, 105, 107, 109

Offset: 1

Eric W. Weisstein

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A022843.

Cf. A108599.

a054385 n = a054385_list !! n
a054385_list = map (floor . (* e') . fromIntegral) [1..]
   where e' = e / (e - 1); e = exp 1
-- Reinhard Zumkeller, Jul 06 2013

A054385 := proc(n)
    floor(n*exp(1)/(exp(1)-1)) ;
end proc: # R. J. Mathar, Jan 25 2015

a[n_] := Floor[n E/(E - 1)];
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

from sympy import E
print([n*E//(E-1) for n in range(1, 70)]) # Karl V. Keller, Jr., Aug 07 2020

A258244 Number of steps from n to 0, where allowable steps are x -> [x/e] if x = is in A022843 (the Beatty sequence for e) and x -> [e*x] otherwise, where [ ] = floor.

Original entry on oeis.org

0, 2, 1, 3, 5, 3, 5, 7, 2, 4, 4, 6, 8, 6, 8, 10, 4, 6, 8, 6, 8, 8, 10, 12, 3, 5, 7, 5, 7, 5, 7, 9, 7, 9, 11, 9, 11, 13, 7, 9, 9, 11, 13, 11, 13, 15, 5, 7, 7, 9, 11, 9, 11, 13, 7, 9, 11, 9, 11, 9, 11, 13, 11, 13, 15, 13, 15, 4, 6, 8, 6, 8, 10, 8, 10, 12, 6, 8

Offset: 0

Clark Kimberling, Jun 08 2015

a(n) = number of edges from 0 to n in the tree at A258243.

			20->54->19->6->16->5->1->2->0, so that a(20) = 8.

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A022843, A258243, A258212.

r = E; w = Table[Floor[r*n], {n, 1, 1000}];
f[x_] := If[MemberQ[w, x], Floor[x/r], Floor[r*x]];
g[x_] := Drop[FixedPointList[f, x], -1];
Table[-1+ Length[g[n]], {n, 0, 100}]

A276859 First differences of the Beatty sequence A022843 for e.

Original entry on oeis.org

2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2

Offset: 1

Clark Kimberling, Sep 24 2016

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A022843, A276875.

A276859:=n->floor(n*exp(1))-floor((n-1)*exp(1)): seq(A276859(n), n=1..200); # Wesley Ivan Hurt, Jan 10 2017

z = 500; r = E; b = Table[Floor[k*r], {k, 0, z}] (* A022843 *)
Differences[b] (* A276859 *)
Differences[Floor[E*Range[0,90]]] (* Harvey P. Dale, Nov 24 2016 *)

a(n) = floor(n*r) - floor(n*r - r), where r = e, n >= 1.

A191455 Dispersion of (floor(n*e)), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 13, 21, 10, 6, 35, 57, 27, 16, 7, 95, 154, 73, 43, 19, 9, 258, 418, 198, 116, 51, 24, 11, 701, 1136, 538, 315, 138, 65, 29, 12, 1905, 3087, 1462, 856, 375, 176, 78, 32, 14, 5178, 8391, 3974, 2326, 1019, 478, 212, 86, 38, 15, 14075, 22809

Offset: 1

Clark Kimberling, Jun 05 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...2....5....13...35
  3...8....21...57...154
  4...10...27...73...198
  6...16...43...116..315
  7...19...51...138..375

Cf. A114537, A035513, A035506.

A191455 := proc(r, c)
    option remember;
    if c = 1 then
        A054385(r) ;
    else
        A022843(procname(r, c-1)) ;
    end if;
end proc: # R. J. Mathar, Jan 25 2015

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=Floor[n*E]   (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191455 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191455 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A276871 Sums-complement of the Beatty sequence for sqrt(5).

Original entry on oeis.org

1, 10, 19, 28, 37, 48, 57, 66, 75, 86, 95, 104, 113, 124, 133, 142, 151, 162, 171, 180, 189, 198, 209, 218, 227, 236, 247, 256, 265, 274, 285, 294, 303, 312, 323, 332, 341, 350, 359, 370, 379, 388, 397, 408, 417, 426, 435, 446, 455, 464, 473, 484, 493, 502

Offset: 1

Clark Kimberling, Sep 24 2016

The sums-complement of a sequence s(1), s(2), ... of positive integers is introduced here as the set of numbers c(1), c(2), ... such that no c(n) is a sum s(j)+s(j+1)+...+s(k) for any j and k satisfying 1 <= j <= k.  If this set is not empty, the term "sums-complement" also applies to the (possibly finite) sequence of numbers c(n) arranged in increasing order.  In particular, the difference sequence D(r) of a Beatty sequence B(r) of an irrational number r > 2 has an infinite sums-complement, abbreviated as SC(r) in the following table:
  r                  B(r)        D(r)       SC(r)
  ----------------------------------------------------
  sqrt(5)            A022839     A081427    A276871
  sqrt(6)            A022840     A276856    A276872
  sqrt(7)            A022841     A276857    A276873
  sqrt(8)            A022842     A276858    A276874
  e                  A022843     A276859    A276875
  2*e                A276853     A276860    A276876
  Pi                 A022844     A063438    A276877
  2*Pi               A028130     A276861    A276878
  1+sqrt(2)          A003151     A276862    A276879
  1+sqrt(3)          A054088     A007538    A276880
  1+sqrt(5)          A276854     A276863    A276881
  2+sqrt(2)          A001952     A276864    A276882
  2+sqrt(3)          A003512     A276865    A276883
  2+sqrt(5)          A004976     A276866    A276884
  1+tau              A001950   2 + A003849  A276885
  2+tau              A003231     A276867    A276886
  3+tau              A276855     A276868    A276887
  2+sqrt(1/2)        A182769     A276869    A276888
  sqrt(2)+sqrt(3)    A110117     A276870    A276889
From Jeffrey Shallit, Aug 15 2023: (Start)
Simpler description:  this sequence represents those positive integers that CANNOT be expressed as a difference of two elements of A022839.
There is a 20-state Fibonacci automaton for the terms of this sequence (see a276871.pdf).  It takes as input the Zeckendorf representation of n and accepts iff n is a member of A276871. (End)

			The Beatty sequence for sqrt(5) is A022839 = (0,2,4,6,8,11,13,15,...), with difference sequence s = A081427 = (2,2,2,2,3,2,2,2,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,4,5,6,7,8,9,11,12,...), with complement (1,10,19,28,37,...).

Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See p. 16.
Jeffrey Shallit, Fibonacci automaton for A276871
Index entries for sequences related to Beatty sequences

Cf. A022839, A081427.

z = 500; r = Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A022839 *)
t = Differences[b]; (* A081427 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w];  (* A276871 *)

A103762 a(n) = least k with Sum_{j = n..k} 1/j >= 1.

Original entry on oeis.org

1, 4, 7, 10, 12, 15, 18, 20, 23, 26, 29, 31, 34, 37, 39, 42, 45, 48, 50, 53, 56, 58, 61, 64, 67, 69, 72, 75, 77, 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

Offset: 1

David W. Wilson, Apr 14 2008

nonn

a(n) = A136617(n) + n for n > 1. Also a(n) = A136616(n-1) + 1 for n > 1.
If you compare this to floor(e*n) = A022843, 2,5,8,10,13,16,..., it appears that floor(e*n)-a(n) = 1,1,1,0,1,1,1,1,1,1,0,..., initially consisting of 0's and 1's. The places where the 0's occur are 4, 11, 18, 25, 32, 36, 43, 50, 57, 64, 71, ... whose differences seem to be 4, 7 or 11.
There are some rather sharp estimates on this type of differences between harmonic numbers in Theorem 3.2 of the Sintamarian reference, which may help to uncover such a pattern. - R. J. Mathar, Apr 15 2008
a(n) = round(e*(n-1/2)) with the exception of the terms of A277603; at those values of n, a(n) = round(e*(n-1/2)) + 1. - Jon E. Schoenfield, Apr 03 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
E. R. Bobo, A sequence related to the harmonic series, College Math. J. 26 (1995), 308-310.
D. T. Clancy and S. J. Kifowit, A closer look at Bobo's sequence, College Math. J. 45 (2014), 199-206.
A. Sintamarian, A generalization of Euler's constant, Numer. Algor. 46 (2007), pp. 141-151.

Cf. A136616, A136617, A242679 (Bobo numbers).

Cf. A081881, A277603, A289183. - Jon E. Schoenfield, Mar 31 2018

i = 0; s = 0; Table[While[s < 1, i++; s = s + 1/i]; s = s - 1/n; i, {n, 100}] (* T. D. Noe, Jun 26 2012 *)

default(realprecision, 10^5); e=exp(1);
a(n) = if(n<2, 1, floor(e*n+(1-e)/2+(e-1/e)/(24*n-12))); \\ Jinyuan Wang, Mar 06 2020

A077545 Primes of the form floor(k*e).

Original entry on oeis.org

2, 5, 13, 19, 29, 43, 59, 67, 73, 89, 97, 103, 127, 149, 157, 163, 173, 179, 233, 239, 241, 263, 269, 271, 277, 293, 307, 331, 337, 347, 353, 383, 421, 443, 467, 521, 557, 587, 617, 619, 641, 701, 709, 733, 739, 761, 769, 823, 829, 839, 853, 883, 907, 929, 937

Offset: 1

Amarnath Murthy, Nov 09 2002

Primes not in A077545 are in A184856, since {floor(k*e)} and {floor(j*e/(e-1))} are complementary Beatty sequences (A022843 and A054385).

Cf. A062409, A184855, A184856, A184587, A184858.

r=E; s=r/(r-1);
a[n_]:=Floor[n*r];
b[n_]:=Floor[n*s];
Table[a[n], {n, 1, 120}]  (* A022843 *)
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1, a[n]]], {n, 1, 600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2, n]], {n, 1, 600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3, n]], {n, 1, 300}]; t3
t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4, b[n]]], {n, 1, 600}]; t4
t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5, n]], {n, 1, 600}]; t5
t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6, n]], {n, 1, 300}]; t6
(* List t1 matches A077545; list t2 matches A062409;
lists t3-t6 match A184855-A184858. *)

More terms from Sascha Kurz, Jan 12 2003

Mathematica code and crossreferences by Clark Kimberling, Jan 24 2011

A022852 Integer nearest n * e, where e is the natural log base.

Original entry on oeis.org

0, 3, 5, 8, 11, 14, 16, 19, 22, 24, 27, 30, 33, 35, 38, 41, 43, 46, 49, 52, 54, 57, 60, 63, 65, 68, 71, 73, 76, 79, 82, 84, 87, 90, 92, 95, 98, 101, 103, 106, 109, 111, 114, 117, 120, 122, 125, 128, 130, 133, 136, 139, 141, 144, 147, 150, 152, 155, 158, 160

Offset: 0

Clark Kimberling

nonn

A022843(n) <= a(n) <= A121384(n). - Reinhard Zumkeller, Mar 17 2015

			2 * e = 5.436563656918..., so a(2) = 5.
3 * e = 8.1548454853771357..., so a(3) = 8.
4 * e = 10.87312731383618..., so a(4) = 11.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A001113, A022843, A121384.

a022852 = round . (* exp 1) . fromIntegral
-- Reinhard Zumkeller, Mar 17 2015

[Round(n*Exp(1)): n in [0..60]]; // G. C. Greubel, Sep 29 2018

Round[E * Range[0, 60]] (* Harvey P. Dale, Jan 08 2014 *)

a(n)=n*exp(1)\1 \\ Charles R Greathouse IV, Jan 08 2014

(0L to 59L).map(n => Math.round(n * Math.E)) // Alonso del Arte, Dec 14 2018

Zero prepended by Harvey P. Dale, Jan 08 2014

A024573 a(n) = floor(1/frac(n*e)).

Original entry on oeis.org

1, 2, 6, 1, 1, 3, 35, 1, 2, 5, 1, 1, 2, 17, 1, 2, 4, 1, 1, 2, 11, 1, 1, 4, 1, 1, 2, 8, 1, 1, 3, 1, 1, 2, 7, 1, 1, 3, 76, 1, 2, 5, 1, 1, 3, 24, 1, 2, 5, 1, 1, 2, 14, 1, 1, 4, 1, 1, 2, 10, 1, 1, 3, 1, 1, 2, 8, 1, 1, 3, 1, 1, 2, 6, 1, 1, 3, 38, 1, 2, 5, 1, 1, 2, 18, 1, 2, 4, 1, 1, 2, 12, 1, 1, 4, 1, 1, 2, 9, 1

Offset: 1

Clark Kimberling

nonn

From Hieronymus Fischer, Apr 15 2012: (Start)
The sequence is well defined, since frac(n*e)>0 for n>0.
Let b(n,m) = |{a(k)| 1<=k<=n, a(k)>=m}| be the number of the first n terms which are >= m >= 1. Then, lim b(n,m)/n = 1/m for n-->oo since frac(n*e) is uniformly distributed. (End)

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A024584, A022843, A000142, A000027.

Cf. A190860, A190847.

seq(floor(1/frac(n*exp(1))), n=1..30); # Ridouane Oudra, Jun 09 2025

f[n_] := Floor[1/FractionalPart[n*E]]; Array[f, 100] (* Robert G. Wilson v, Apr 17 2012 *)

From Ridouane Oudra, Jun 09 2025: (Start)
a(n!) = n.
a(n) = 1 if n is in A190860.
a(n) > 1 if n is in A190847. (End)

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 *)

cp's OEIS Frontend

A054385 Beatty sequence for e/(e-1); complement of A022843.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

A258244 Number of steps from n to 0, where allowable steps are x -> [x/e] if x = is in A022843 (the Beatty sequence for e) and x -> [e*x] otherwise, where [ ] = floor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A276859 First differences of the Beatty sequence A022843 for e.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A191455 Dispersion of (floor(n*e)), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A276871 Sums-complement of the Beatty sequence for sqrt(5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A103762 a(n) = least k with Sum_{j = n..k} 1/j >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A077545 Primes of the form floor(k*e).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A022852 Integer nearest n * e, where e is the natural log base.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs