A184799 -id:A184799 - 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

A184796 Primes of the form floor(k*sqrt(3)).

Original entry on oeis.org

3, 5, 13, 17, 19, 29, 31, 41, 43, 53, 67, 71, 79, 83, 103, 107, 109, 131, 157, 173, 181, 193, 197, 199, 211, 223, 233, 239, 251, 263, 271, 277, 311, 313, 337, 349, 353, 367, 379, 389, 401, 419, 431, 433, 439, 443, 457, 467, 479, 491, 509, 521, 523, 547, 557, 569, 571, 587, 599, 601, 607, 613, 647, 659, 661, 673, 677, 691, 701, 727, 739, 743, 751, 769, 827, 829, 853, 857, 859, 881, 883, 907, 911, 919, 937, 947, 971, 983, 997, 1009, 1013, 1021, 1039

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

See A184774.

Equals the prime terms of A022838. - Bill McEachen, Oct 28 2021

			The sequence A022838(n)=floor(n*sqrt(3)) begins with 1,3,5,6,8,10,12,13,15,17,19,... which includes the primes A022838(2)=3, A022838(3)=5, A022838(8)=13,...

Cf. A184774, A184797, A184798, A184799, A184800, A184801.

Cf. A022838. - Bill McEachen, Oct 28 2021

r=3^(1/2); s=r/(r-1);
a[n_]:=Floor [n*r];  (* A022838 *)
b[n_]:=Floor [n*s];  (* A054406 *)
Table[a[n],{n,1,120}]
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
(* The lists t1, t2, t3, t4, t5, t6 match the sequences
A184796, A184797, A184798, A184799, A184800, A184801. *)

A184801 Numbers m such that prime(m) is of the form floor(ks), where s=(3+sqrt(3))/2; complement of A184778.

Original entry on oeis.org

1, 4, 5, 9, 12, 15, 17, 18, 21, 24, 25, 26, 30, 31, 33, 34, 35, 36, 38, 39, 41, 43, 49, 50, 53, 55, 57, 60, 61, 62, 63, 66, 67, 69, 72, 74, 76, 78, 80, 82, 87, 89, 90, 93, 95, 96, 100, 103, 106, 108, 113, 114, 115, 116, 117, 119, 124, 127, 128, 130, 134, 135, 137, 138, 139, 140, 141, 142, 143, 146, 150, 151, 154, 158, 160, 162, 163, 165, 167, 171, 173, 174, 180, 183, 184, 186, 188, 193, 195, 197, 198, 200, 203, 204, 206, 209, 210, 212, 213, 217, 219, 221

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

			See A184796.

A184796, A184797, A184799, A184800.

Mathematica
```
(See A184796.)
```

A184800 Numbers k such that floor(k*s) is prime, where s = (3 + sqrt(3))/2.

Original entry on oeis.org

1, 3, 5, 10, 16, 20, 25, 26, 31, 38, 41, 43, 48, 54, 58, 59, 63, 64, 69, 71, 76, 81, 96, 97, 102, 109, 114, 119, 120, 124, 130, 134, 140, 147, 152, 158, 162, 168, 173, 178, 190, 195, 196, 206, 211, 213, 229, 238, 244, 251, 261, 262, 267, 271, 272, 276, 289, 300, 304, 310, 320, 322, 327, 333, 337, 342, 343, 347, 348, 355, 365, 371, 375, 393, 398, 403, 409, 413, 419, 431, 436, 437, 452, 462, 464, 469, 475, 495, 502, 508, 513, 517, 523, 528, 540, 545, 546, 550, 551, 561, 578, 584

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

			See A184796.

Cf. A054406, A184796, A184799, A184801.

Mathematica
```
(See A184796.)
```

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

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A184796 Primes of the form floor(k*sqrt(3)).

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A184801 Numbers m such that prime(m) is of the form floor(ks), where s=(3+sqrt(3))/2; complement of A184778.

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A184800 Numbers k such that floor(k*s) is prime, where s = (3 + sqrt(3))/2.

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