A184796 -id:A184796 - cp's OEIS frontend

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

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 38, 39, 41, 43, 45, 46, 48, 50, 51, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 90, 91, 93, 95, 96, 98, 100, 102, 103, 105, 107, 109, 110, 112

Offset: 1

Clark Kimberling

nonn

0 <= A144077(n) - a(n) <= 1. - Reinhard Zumkeller, Sep 09 2008
From Reinhard Zumkeller, Jan 20 2010: (Start)
A080757(n) = a(n+1) - a(n).
A171970(n) = floor(a(n)/2).
A171972(n) = a(A000290(n)). (End)
Numbers k>0 such that A194979(k+1) = A194979(k) + 1. - Clark Kimberling, Dec 02 2014
Powers of 2 (i.e, 1, 8, 32, 64, 128, 256, 512, 4096, 8192,...) appear at n=1, 5, 19, 37, 74, 148, 296, 2365, 4730, 18919, 75675, 151349, 302698, 605396, ... related to A293328. - R. J. Mathar, Jan 17 2025

Reinhard Zumkeller, 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. A080757 (first differences), A194106 (partial sums), A194028 (even bisection), A184796 (prime terms).

Cf. A026255, A054406 (complement).

a022838 = floor . (* sqrt 3) . fromIntegral
-- Reinhard Zumkeller, Sep 14 2014

[Floor(n*Sqrt(3)): n in [1..60]]; // G. C. Greubel, Sep 28 2018

A022838 := proc(n)
    floor(n*sqrt(3)) ;
end proc: # R. J. Mathar, Mar 25 2013

Table[Floor[n 3^(1/2)] , {n, 1, 65}] (* Geoffrey Critzer, Jan 11 2015 *)

vector(60, n, floor(n*sqrt(3))) \\ G. C. Greubel, Sep 28 2018

a(n)=sqrtint(3*n^2) \\ Charles R Greathouse IV, Nov 01 2021

from math import isqrt
def A022838(n): return isqrt(3*n*n) # Chai Wah Wu, Aug 06 2022

a(n) = floor(n*sqrt(3)). - Reinhard Zumkeller, Jan 20 2010

a(n) = 2 * floor(n * (sqrt(3) - 1)) + floor(n * (2 - sqrt(3))) + 1. - Miko Labalan, Dec 03 2016

A184799 Primes of the form floor(k*s), where s=(3+sqrt(3))/2.

Original entry on oeis.org

2, 7, 11, 23, 37, 47, 59, 61, 73, 89, 97, 101, 113, 127, 137, 139, 149, 151, 163, 167, 179, 191, 227, 229, 241, 257, 269, 281, 283, 293, 307, 317, 331, 347, 359, 373, 383, 397, 409, 421, 449, 461, 463, 487, 499, 503, 541, 563, 577, 593, 617, 619, 631, 641, 643, 653, 683, 709, 719, 733, 757, 761

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

			See A184796.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A184796, A184800, A184801.

filter:= proc(p) local v, k;
  v:= p*(1-1/sqrt(3));
  k:= ceil(v);
  is((v-k+1)^2 > 1/3)
end proc:
select(filter, [seq(ithprime(i),i=1..200)]); # Robert Israel, May 04 2020

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

A095280 Lower Wythoff primes, i.e., primes in A000201.

Original entry on oeis.org

3, 11, 17, 19, 29, 37, 43, 53, 59, 61, 67, 71, 79, 97, 101, 103, 113, 127, 131, 137, 139, 163, 173, 179, 181, 197, 199, 211, 223, 229, 239, 241, 257, 263, 271, 281, 283, 307, 313, 317, 331, 347, 349, 359, 367, 373, 383, 389, 401, 409, 419, 433

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Contains all primes p whose Zeckendorf-expansion A014417(p) ends with an even number of 0's.

For generalizations and conjectures, see A184774.

Robert Israel, Table of n, a(n) for n = 1..10000
Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 & A000201. Complement of A095281 in A000040. Cf. A095080, A095083, A095084, A095290, A184792, A184793, A184794, A184796.

R:= NULL: count:= 0:
for n from 1 while count < 100 do
  p:= floor(n*phi);
  if isprime(p) then R:= R,p; count:= count+1 fi
od:
R; # Robert Israel, Jan 17 2023

Mathematica
```
(See A184792.)
```

from math import isqrt
from itertools import count, islice
from sympy import isprime
def A095280_gen(): # generator of terms
    return filter(isprime,((n+isqrt(5*n**2)>>1) for n in count(1)))
A095280_list = list(islice(A095280_gen(),30)) # Chai Wah Wu, Aug 16 2022

A184797 Numbers k such that floor(k*sqrt(3)) is prime.

Original entry on oeis.org

2, 3, 8, 10, 11, 17, 18, 24, 25, 31, 39, 41, 46, 48, 60, 62, 63, 76, 91, 100, 105, 112, 114, 115, 122, 129, 135, 138, 145, 152, 157, 160, 180, 181, 195, 202, 204, 212, 219, 225, 232, 242, 249, 250, 254, 256, 264, 270, 277, 284, 294, 301, 302, 316, 322, 329, 330, 339, 346, 347, 351, 354, 374, 381, 382, 389, 391, 399, 405, 420, 427, 429, 434, 444, 478, 479, 493, 495, 496, 509, 510, 524, 526, 531, 541, 547, 561, 568, 576, 583, 585, 590, 600

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

			See A184796.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002194, A022838, A184796, A184798.

(See A184796.)
Select[Range[600],PrimeQ[Floor[# Sqrt[3]]]&] (* Harvey P. Dale, May 21 2017 *)

is(n)=isprime(sqrtint(3*n^2)) \\ Charles R Greathouse IV, May 22 2017

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.)
```

A323388 a(n) = b(n+1)/b(n) - 1 where b(1)=3 and b(k) = b(k-1) + lcm(floor(sqrt(3)*k), b(k-1)).

Original entry on oeis.org

1, 5, 1, 1, 5, 1, 13, 5, 17, 19, 1, 11, 1, 5, 1, 29, 31, 1, 17, 1, 19, 13, 41, 43, 1, 23, 1, 1, 17, 53, 1, 19, 29, 1, 31, 1, 13, 67, 23, 71, 1, 37, 1, 1, 79, 1, 83, 1, 43, 1, 1, 13, 31, 1, 1, 1, 1, 1, 103, 1, 107, 109, 1, 1, 1, 29, 1, 1, 1, 61, 1, 1

Offset: 1

Pedja Terzic, Jan 13 2019

nonn

Conjectures:
1. This sequence consists only of 1's and primes.
2. Every odd prime of the form floor(sqrt(3)*m) greater than 3 is a term of this sequence.
3. At the first appearance of each prime of the form floor(sqrt(3)*m), it is larger than any prime that has already appeared.
The 2nd and 3rd conjectures are proved at the Mathematics Stack Exchange link. - Sungjin Kim, Jul 17 2019

Michel Marcus, Table of n, a(n) for n = 1..10000
Mathematics Stack Exchange, Generating primes of the form floor(sqrt(3)*n)

Cf. A184796 (primes of the form floor(sqrt(3)*m)).

Cf. A135506, A323359, A323386.

Generator(n)={b1=3; list=[]; for(k=2, n, b2=b1+lcm(floor(sqrt(3)*k), b1); a=b2/b1-1; list=concat(list,a); b1=b2); print(list)}

lista(nn)={my(b1=3, b2, va=vector(nn)); for(k=2, nn+1, b2=b1+lcm(sqrtint(3*k^2), b1); va[k-1]=b2/b1-1; b1=b2); va}; \\ Michel Marcus, Aug 20 2022

A184798 Numbers m such that prime(m) is of the form floor(k*sqrt(3)); complement of A184801.

Original entry on oeis.org

2, 3, 6, 7, 8, 10, 11, 13, 14, 16, 19, 20, 22, 23, 27, 28, 29, 32, 37, 40, 42, 44, 45, 46, 47, 48, 51, 52, 54, 56, 58, 59, 64, 65, 68, 70, 71, 73, 75, 77, 79, 81, 83, 84, 85, 86, 88, 91, 92, 94, 97, 98, 99, 101, 102, 104, 105, 107, 109, 110, 111, 112, 118, 120, 121, 122, 123, 125, 126, 129, 131, 132, 133, 136, 144, 145, 147, 148, 149, 152, 153, 155, 156, 157, 159, 161, 164, 166, 168, 169, 170, 172, 175

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

			See A184796.

Cf. A184796, A184797, A184801.

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

cp's OEIS Frontend

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

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A184799 Primes of the form floor(k*s), where s=(3+sqrt(3))/2.

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A095280 Lower Wythoff primes, i.e., primes in A000201.

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A184797 Numbers k such that floor(k*sqrt(3)) is prime.

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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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Mathematica

A323388 a(n) = b(n+1)/b(n) - 1 where b(1)=3 and b(k) = b(k-1) + lcm(floor(sqrt(3)*k), b(k-1)).

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PARI

A184798 Numbers m such that prime(m) is of the form floor(k*sqrt(3)); complement of A184801.

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