A054385 -id:A054385 - cp's OEIS frontend

A245223 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.

3, 6, 9, 3, 0, 6, 3, 9, 6, 4, 5, 3, 0, 1, 2, 3, 0, 5, 9, 7, 2, 7, 8, 1, 6, 9, 3, 6, 8, 7, 1, 9, 0, 6, 6, 9, 4, 4, 5, 6, 3, 1, 3, 3, 1, 6, 9, 0, 3, 8, 4, 9, 6, 0, 5, 3, 1, 0, 9, 1, 0, 0, 2, 8, 8, 6, 3, 3, 4, 6, 9, 2, 4, 5, 3, 0, 2, 7, 0, 1, 2, 6, 2, 9, 8, 0

Offset: 1

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 0.36930639645301230597278169368719066944...  The first 16 numbers f(n,1) comprise S(16) = {1, 2, 1/2, 3/2, 5/2, 2/5, 7/5, 12/5, 5/12, 17/12, 12/17, 29/17}; min(S(16)) = 17/46 = 0.36956... and max(S(12)) = 46/17 = 2.7058...

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

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245220, A245224.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = E/(E-1); w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; $RecursionLimit = tmpRec;
m = Min[N[Table[s[n], {n, 1, 4000}], 300]]
RealDigits[m]  (* A245223 *)
(* Peter J. C. Moses, Jul 04 2014 *)

a(n)*sup{f(n,1)} = 1.

A245224 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.

Original entry on oeis.org

2, 7, 0, 7, 7, 7, 8, 7, 1, 6, 0, 0, 5, 0, 7, 8, 1, 2, 4, 3, 4, 0, 2, 0, 6, 6, 6, 5, 9, 6, 3, 1, 3, 1, 6, 2, 9, 9, 2, 3, 3, 1, 2, 4, 2, 4, 9, 1, 0, 4, 4, 5, 1, 7, 6, 6, 6, 9, 1, 3, 7, 9, 1, 8, 3, 4, 6, 4, 8, 3, 0, 8, 8, 4, 3, 2, 3, 4, 7, 0, 0, 2, 3, 5, 5, 3

Offset: 1

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 2.7077787160050781243402066659631316299233...  The first 16 numbers f(n,1) comprise S(16) = {1, 2, 1/2, 3/2, 5/2, 2/5, 7/5, 12/5, 5/12, 17/12, 12/17, 29/17}; min(S(16)) = 17/46 = 0.36956... and max(S(12)) = 46/17 = 2.7058...

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

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245220, A245224.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = E/(E-1); w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; $RecursionLimit = tmpRec;
m = Max[N[Table[s[n], {n, 1, 4000}], 300]]
RealDigits[m]  (* A245224 *)
(* Peter J. C. Moses, Jul 04 2014 *)

a(n)*inf{f(n,1)} = 1.

A245225 Continued fraction expansion of the constant c in A245224; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.

Original entry on oeis.org

2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2

Offset: 0

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 2.70777871600507812434020666596313162... ; The first 16 numbers f(n,1) comprise S(16) = {1, 2, 1/2, 3/2, 5/2, 2/5, 7/5, 12/5, 5/12, 17/12, 12/17, 29/17}; max(S(16)) = 46/17, with continued fraction [2, 1, 2, 2, 2].

Cf. A226080 (infinite Fibonacci tree), A245217, A245219, A245222, A245224 (decimal expansion).

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = E/(E-1); w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; max = Max[N[Table[s[n], {n, 1, 3000}], 200]] (* A245224 *)
ContinuedFraction[max, 120] (* A245225 *)

Offset changed by Andrew Howroyd, Jul 07 2024

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 *)

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

A245215 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = f(n-1,x) + 1 if n is in A000201, else f(n,x) = 1/f(n-1,x).

Original entry on oeis.org

3, 6, 6, 3, 0, 4, 6, 9, 4, 6, 5, 3, 2, 7, 2, 6, 5, 6, 6, 8, 2, 4, 9, 4, 1, 3, 1, 4, 2, 9, 0, 9, 6, 6, 9, 2, 9, 9, 8, 4, 2, 7, 8, 8, 9, 3, 9, 2, 5, 4, 3, 1, 6, 0, 4, 1, 0, 3, 1, 0, 3, 8, 0, 6, 3, 6, 0, 0, 5, 6, 4, 5, 2, 9, 0, 6, 1, 5, 4, 6, 1, 6, 9, 4, 9, 5

Offset: 1

Clark Kimberling, Jul 13 2014

Equivalently, f(n,x) = 1/(f(n-1,x) if n is in A001950 (upper Wythoff sequence, given by w(n) = floor[tau*n], where tau = (1 + sqrt(5))/2, the golden ratio) and f(n,x) = f(n-1) + 1 otherwise.  Let c = inf{f(n,1)}.  The continued fraction of c is [0,2,1,2,1,2,2,1,2,2,1,2, ...], and the continued fraction of sup{f(n,x)}, alias -2 + 1/c, appears to be identical to the Hofstadter eta-sequence at A006340:  (2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2,...).  Other limiting constants are similarly obtained using other pairs of Beatty sequences:
...
Beatty sequence .... inf{f(n,1)} ... sup{f(n,1)}
A000201 (tau) ...... A245215 ....... A245216
A001951 (sqrt(2)) .. A245217 ....... A245218; cont. fr. A245219
A022838 (sqrt(3)) .. A245220 ....... A245221; cont. fr. A245222
A054385 (e/(e-1)) .. A245223 ....... A245224; cont. fr. A245225

			c = 0.366304694653272656682494131429096692998...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 1/2, 3/2, 5/2, 2/5, 7/5, 5/7, 12/7, 19/7, 7/19, 26/19}; min(S(12)) = 7/19 = 0.36842...

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

Cf. A226080 (infinite Fibonacci tree), A006340, A245216, A245217, A245220, A245223, A246129.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = GoldenRatio; w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; $RecursionLimit = tmpRec;
m = Min[N[Table[s[n], {n, 1, 4000}], 300]]
t = RealDigits[m]  (* A245215 *)
(* Peter J. C. Moses, Jul 04 2014 *)

a(n)*(2 + sup{f(n,1)}) = 1.

Equals 1/A245216 = A246129 - 2. - Hugo Pfoertner, Nov 10 2024

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

A108599 Self-inverse integer permutation induced by Beatty sequences for e and e/(e-1).

Original entry on oeis.org

2, 1, 5, 8, 3, 10, 13, 4, 16, 6, 19, 21, 7, 24, 27, 9, 29, 32, 11, 35, 12, 38, 40, 14, 43, 46, 15, 48, 17, 51, 54, 18, 57, 59, 20, 62, 65, 22, 67, 23, 70, 73, 25, 76, 78, 26, 81, 28, 84, 86, 30, 89, 92, 31, 95, 97, 33, 100, 34, 103, 106, 36, 108, 111, 37, 114, 39, 116, 119, 41

Offset: 1

Reinhard Zumkeller, Jun 11 2005

nonn

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to Beatty sequences

Cf. A108591.

a(A022843(n))=A054385(n) and a(A054385(n))=A022843(n).

A184856 Primes of the form floor(k*e/(e-1)).

Original entry on oeis.org

3, 7, 11, 17, 23, 31, 37, 41, 47, 53, 61, 71, 79, 83, 101, 107, 109, 113, 131, 137, 139, 151, 167, 181, 191, 193, 197, 199, 211, 223, 227, 229, 251, 257, 281, 283, 311, 313, 317, 349, 359, 367, 373, 379, 389, 397, 401, 409, 419, 431, 433, 439, 449, 457, 461, 463, 479, 487, 491, 499, 503, 509, 523, 541, 547, 563, 569, 571, 577, 593, 599, 601, 607, 613, 631, 643, 647, 653, 659, 661, 673, 677, 683, 691, 719, 727, 743, 751, 757, 773, 787, 797, 809, 811, 821, 827, 857, 859, 863, 877, 881, 887, 911, 919, 941, 947

Offset: 1

Clark Kimberling, Jan 23 2011

nonn

Primes in A054385. - Bill McEachen, Nov 04 2021

			See A077545.

Cf. A077545, A184855, A054385.

Mathematica
```
(See A077545.)
```

select( {is_A184856(n, c=1-exp(-1)) = isprime(n) && for(k=floor(n*c),ceil((n+1)*c), k\c==n && return(1))}, [1..999]) \\ M. F. Hasler, Jul 12 2024

Complement of A077545 in the primes: A000040 \ A077545. - M. F. Hasler, Jul 12 2024

A325749 First term of n-th difference sequence of (floor(e*k/(e-1))), k >= 0.

Original entry on oeis.org

1, 1, -2, 4, -8, 16, -31, 56, -91, 126, -126, 0, 462, -1715, 4704, -11319, 25186, -52920, 105841, -201704, 364819, -620312, 973028, -1352078, 1486674, -657799, -2750022, 11967739, -32805248, 74480266, -148960531, 265105676, -408608551, 487599186, -201501426

Offset: 1

Clark Kimberling, Jun 07 2019

Clark Kimberling, Table of n, a(n) for n = 1..200

Cf. A054385.

Table[First[Differences[Table[Floor[n*E/(E-1)], {n, 0, 50}], n]], {n, 1, 50}]

cp's OEIS Frontend

A245223 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.

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A245224 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.

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A245225 Continued fraction expansion of the constant c in A245224; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.

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

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Maple

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

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A245215 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = f(n-1,x) + 1 if n is in A000201, else f(n,x) = 1/f(n-1,x).

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Mathematica

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

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Mathematica

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A108599 Self-inverse integer permutation induced by Beatty sequences for e and e/(e-1).

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