A022838 -id:A022838 - cp's OEIS frontend

A054406 Beatty sequence for (3+sqrt 3)/2; complement of A022838.

2, 4, 7, 9, 11, 14, 16, 18, 21, 23, 26, 28, 30, 33, 35, 37, 40, 42, 44, 47, 49, 52, 54, 56, 59, 61, 63, 66, 68, 70, 73, 75, 78, 80, 82, 85, 87, 89, 92, 94, 97, 99, 101, 104, 106, 108, 111, 113, 115, 118, 120, 123, 125, 127, 130, 132, 134, 137, 139, 141, 144, 146

Offset: 1

Eric W. Weisstein

nonn

Numbers k such that A194979(k+1) = A194979(k). - Clark Kimberling, Dec 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A194143 (partial sums), A182778 (even bisection), A184799 (prime terms).
Cf. A022838 (complement), A026255.
Cf. A194979.

[Floor(n*(3+Sqrt(3))/2): n in [1..70]]; // Vincenzo Librandi, Oct 25 2011

A054406 := proc(n) n*(3+sqrt(3))/2 ; floor(%) ; end proc: # R. J. Mathar, Feb 26 2011

a054406[n_Integer] := Floor[# (3 + Sqrt[3])/2] & /@ Range[n]; a054406[62] (* Michael De Vlieger, Dec 14 2014 *)

is(n)=sqrtint((n+1)^2\3)==sqrtint(n^2\3) \\ Charles R Greathouse IV, Nov 01 2021

A356088 a(n) = A001951(A022838(n)).

Original entry on oeis.org

1, 4, 7, 8, 11, 14, 16, 18, 21, 24, 26, 28, 31, 33, 35, 38, 41, 43, 45, 48, 50, 53, 55, 57, 60, 63, 65, 67, 70, 72, 74, 77, 80, 82, 84, 87, 90, 91, 94, 97, 100, 101, 104, 107, 108, 111, 114, 117, 118, 121, 124, 127, 128, 131, 134, 135, 138, 141, 144, 145

Offset: 1

Clark Kimberling, Aug 04 2022

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) u' o v'.
Every positive integer is in exactly one of the four sequences.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356088, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3)), so that r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2.

			(1)  u o v   = (1,  4,  7,  8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088.
(2)  u o v'  = (2,  5,  9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089.
(3)  u' o v  = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090.
(4)  u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091.

Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356089, A356090, A356091.

z = 600; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];  (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[u[[v[[n]]]], {n, 1, zz}];    (* A356088 *)
Table[u[[v1[[n]]]], {n, 1, zz}];   (* A356089 *)
Table[u1[[v[[n]]]], {n, 1, zz}];   (* A356090 *)
Table[u1[[v1[[n]]]], {n, 1, zz}];  (* A356091 *)

from math import isqrt
def A356088(n): return isqrt(isqrt(3*n*n)**2<<1) # Chai Wah Wu, Aug 06 2022

A245222 Continued fraction expansion of the constant c in A245221; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jul 14 2014

See Comments at A245215.

Appears to be the same as the sequence 1 + [x == 0 (mod sqrt(3))], as x runs over the elements of N U N*sqrt(3) in increasing order, where N = {0, 1, 2, 3, ...} and [...] is the Iverson bracket. - M. F. Hasler, Feb 06 2025

			c = 2.7207664507294752975469517348171513242... ; the first 12 numbers f(n,1) comprise S(12) = {1, 2, 1/2, 3/2, 2/3, 5/3, 8/3, 3/8, 11/8, 8/11, 19/11, 11/19}; max(S(12)) = 8/3, with continued fraction [2,1,2].
From _M. F. Hasler_, Feb 06 2025: (Start)
Illustration of the "multiple of sqrt(3)" comment:
  n:  0   1   2    3   4    5    6   7   8    9   10  11  12   13  14  15
  x:  0   1  1.73  2   3   3.46  4   5  5.20  6  6.93  7   8  8.66  9  10
  m:  1   0   1    0   0    1    0   0   1    0   1    0   0   1    0   0
Here, x lists the elements of N U N*sqrt(3), and m = 1 if x == 0 (mod sqrt(3)), i.e., x is an integer multiple of sqrt(3). The sequence a(n) is m + 1. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1999

Cf. A226080 (infinite Fibonacci tree), A245217, A245219, A245220, A245221 (decimal expansion).
Cf. A144612 (Sturmian word of slope (3-sqrt(3))/2; same as 2-a(n)).
Cf. A006337 (Hofstadter's eta sequence: an analog with sqrt(2)).

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[3]; 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]] (* A245221 *)
ContinuedFraction[max, 120] (* A245222 *)

/* illustration of the comment related to sqrt(3)*/
[1+(abs(x-x\/s*s)<1e-9) | x<-Set(concat(Col([1.,s=sqrt(3)]~*[0..99])))[1..99] ] \\ M. F. Hasler, Feb 06 2025

a(n) = 2 - A144612(n) for all n > 0. - M. F. Hasler, Feb 06 2025

Offset changed by Andrew Howroyd, Aug 08 2024

A245220 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 0.367543491184951248721260972541092540...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 1/2, 3/2, 2/3, 5/3, 8/3, 3/8, 11/8, 8/11, 19/11, 11/19}; min(S(12)) = 3/8 = 0.375... and max(S(12)) = 8/3 = 2.666...

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

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245221, A245222.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[3]; 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]  (* A245220 *)
(* Peter J. C. Moses, Jul 04 2014 *)

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

A191336 (A022838 mod 2)+(A054406 mod 2).

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 2, 1, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 2, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 2, 1, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0

Offset: 1

Clark Kimberling, Jun 01 2011

nonn

A022838: Beatty sequence for r=sqrt(3),
A054406: Beatty sequence for s=(3+sqrt(3))/2 (complement
of A022838), so that
A191336(n)=([nr] mod 2)+([ns] mod 2), where [ ]=floor.
A191336(n)=(number of odd numbers in {[nr],[ns]}).

Cf. A191329, A191337, A191338, A191339.

r = Sqrt[3]; s = r/(r - 1); h = 320;
u = Table[Floor[n*r], {n, 1, h}] (* A022838 *)
v = Table[Floor[n*s], {n, 1, h}] (* A054406 *)
w = Mod[u, 2] + Mod[v, 2] (* A191336 *)
Flatten[Position[w, 0]]   (* A191337 *)
Flatten[Position[w, 1]]   (* A191338 *)
Flatten[Position[w, 2]]   (* A191339 *)

a(n)=([nr] mod 2)+([ns] mod 2), where r=sqrt(3), s=r/(r-1), and [ ]=floor.

A245221 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 0.367543491184951248721260972541092540...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 1/2, 3/2, 2/3, 5/3, 8/3, 3/8, 11/8, 8/11, 19/11, 11/19}; min(S(12)) = 3/8 = 0.375... and max(S(12)) = 8/3 = 2.666...

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

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245220, A245222.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[3]; 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]]
t = RealDigits[m]  (* A245221 *)
(* Peter J. C. Moses, Jul 04 2014 *)

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

A356086 Intersection of A001952 and A022838.

Original entry on oeis.org

3, 6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, 81, 88, 95, 102, 105, 109, 112, 116, 119, 122, 126, 129, 133, 136, 143, 150, 157, 174, 180, 187, 204, 211, 218, 221, 225, 228, 232, 235, 242, 245, 249, 252, 256, 259, 266, 273, 284, 285, 287, 289, 290, 292, 294

Offset: 1

Clark Kimberling, Aug 04 2022

This is the third of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A346308.

			(1)  u ^ v   = ( 1,  5,  8, 12, 15, 19, 22, 24, 25, 29, 31, 32, ...) = A346308
(2)  u ^ v'  = ( 2,  4,  7,  9, 11, 14, 16, 18, 21, 26, 28, 33, ...) = A356085
(3)  u' ^ v  = ( 3,  6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, ...) = A356086
(4)  u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087

Cf. A001951, A001952, A022838, A054406, A346308, A356085, A356087, A356088 (results of composition instead of intersections).

z = 200;
r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
r1 = Sqrt[3]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022838 *)
v1 = Take[Complement[Range[1000], v], z]  (* A054406 *)
Intersection[u, v]    (* A346308 *)
Intersection[u, v1]   (* A356085 *)
Intersection[u1, v]   (* A356086 *)
Intersection[u1, v1]  (* A356087 *)

from math import isqrt
from itertools import count, islice
def A356086_gen(): # generator of terms
    return filter(lambda n:n == isqrt(3*(isqrt(n**2//3)+1)**2),((k:=n<<1)+isqrt(k*n) for n in count(1)))
A356086_list = list(islice(A356086_gen(),30)) # Chai Wah Wu, Aug 06 2022

A356090 a(n) = A001952(A022838(n)).

Original entry on oeis.org

3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, 68, 75, 81, 85, 92, 99, 105, 109, 116, 122, 129, 133, 139, 146, 153, 157, 163, 170, 174, 180, 187, 194, 198, 204, 211, 218, 221, 228, 235, 242, 245, 252, 259, 262, 269, 276, 283, 286, 293, 300, 307, 310, 317, 324

Offset: 1

Clark Kimberling, Aug 04 2022

This is the third of four sequences that partition the positive integers. See A356088.

			(1)  u o v   = (1,  4,  7,  8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088
(2)  u o v'  = (2,  5,  9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089
(3)  u' o v  = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090
(4)  u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091

Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356088, A356089, A356091.

z = 600; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];  (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356088 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356089 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356090 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356091 *)

A356180 a(n) = A022838(A001951(n)).

Original entry on oeis.org

1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, 36, 38, 41, 43, 45, 48, 50, 53, 55, 57, 60, 62, 65, 67, 71, 72, 74, 77, 79, 83, 84, 86, 90, 91, 95, 96, 98, 102, 103, 107, 109, 112, 114, 116, 119, 121, 124, 126, 128, 131, 133, 136, 138, 142, 143, 145, 148

Offset: 1

Clark Kimberling, Aug 24 2022

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) v o u, defined by (v o u)(n)  = v(u(n));
(2) v' o u;
(3) v o u';
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. For the reverse composites, u o v, u o v', u' o v, u' o v', see A356088 to A356091.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w.  For w = u,v,u',v', denote the densities by r,s,r',s'.  Then the densities of sequences (1)-(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356180, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3)), so r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2.

			(1)  v o u = (1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, ...) = A356180
(2)  v' o u = (2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, ...) = A356181
(3)  v o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, ...) = A356182
(4)  v' o u' = (7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, ...) = A356183

Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356181, A356182, A356183.

z = 800; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];       (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[v[[u[[n]]]], {n, 1, zz}]      (* A356180 *)
Table[v1[[u[[n]]]], {n, 1, zz}]     (* A356181 *)
Table[v[[u1[[n]]]], {n, 1, zz}]     (* A356182 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A356183 *)

from math import isqrt
def A356180(n): return isqrt(3*isqrt(n**2<<1)**2) # Chai Wah Wu, Sep 06 2022

A356182 a(n) = A022838(A001952(n)).

Original entry on oeis.org

5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, 81, 88, 93, 100, 105, 110, 117, 122, 129, 135, 140, 147, 152, 159, 164, 171, 176, 181, 188, 193, 200, 206, 211, 218, 223, 230, 235, 240, 247, 252, 259, 265, 271, 277, 282, 289, 294, 301, 306, 311, 318, 323

Offset: 1

Clark Kimberling, Aug 24 2022

This is the third of four sequences that partition the positive integers. See A356180.

			(1)  v o u = (1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, ...) = A356180
(2)  v' o u = (2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, ...) = A356181
(3)  v o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, ...) = A356182
(4)  v' o u' = (7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, ...) = A356183

Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356180, A356181, A356183.

z = 800; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];       (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[v[[u[[n]]]], {n, 1, zz}]      (* A356180 *)
Table[v1[[u[[n]]]], {n, 1, zz}]     (* A356181 *)
Table[v[[u1[[n]]]], {n, 1, zz}]     (* A356182 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A356183 *)

from math import isqrt
def A356182(n): return isqrt(3*((k:=n<<1)+isqrt(k*n))**2) # Chai Wah Wu, Sep 05 2022

cp's OEIS Frontend

A054406 Beatty sequence for (3+sqrt 3)/2; complement of A022838.

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A356088 a(n) = A001951(A022838(n)).

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A245222 Continued fraction expansion of the constant c in A245221; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

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A245220 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

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A191336 (A022838 mod 2)+(A054406 mod 2).

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A245221 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

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A356086 Intersection of A001952 and A022838.

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A356090 a(n) = A001952(A022838(n)).

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