A184793 -id:A184793 - cp's OEIS frontend

A184795 Numbers m such that prime(m) is of the form floor(k*s), where s=(3+sqrt(5))/2; complement of A184793.

1, 3, 4, 6, 9, 11, 13, 15, 21, 23, 24, 28, 29, 35, 36, 37, 39, 43, 44, 49, 51, 54, 57, 59, 62, 64, 68, 71, 75, 78, 82, 83, 85, 90, 92, 99, 100, 101, 102, 109, 111, 113, 116, 120, 124, 125, 126, 127, 128, 129, 130, 132, 133, 135, 136, 141, 144, 145, 147, 150, 151, 154, 156, 158, 159, 161, 164, 168, 170, 172, 173, 175

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

Indices of primes that appear in the Beatty sequence A001950, the Beatty sequence of A104457. - R. J. Mathar, Jan 22 2011

			Prime(1)=2 with k=1. Prime(3)=5 with k=2. Prime(4)=7 with k=3.

Cf. A184774, A184792, A184793.

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

A184792 Numbers k such that floor(k*r) is prime, where r = golden ratio=(1+sqrt(5))/2.

Original entry on oeis.org

2, 7, 11, 12, 18, 23, 27, 33, 37, 38, 42, 44, 49, 60, 63, 64, 70, 79, 81, 85, 86, 101, 107, 111, 112, 122, 123, 131, 138, 142, 148, 149, 159, 163, 168, 174, 175, 190, 194, 196, 205, 215, 216, 222, 227, 231, 237, 241, 248, 253, 259, 268, 274, 278, 283, 285, 289, 301, 304, 309, 311, 315, 322, 348, 352, 353, 357, 363, 367, 372, 379, 383, 390, 398, 400, 404, 409, 416, 419, 457, 468, 478, 487, 493, 500, 508, 509, 519, 530, 531, 545, 546, 561, 568, 582, 589, 598

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

			The sequence L(n)=floor(n*r) begins with
1,3,4,6,8,9,11,12,14,16,17,...,
which includes the primes L(2)=3, L(7)=11,...

Cf. A184774, A095280, A184792, A184793, A095281, A184794, A184795.

r=(1+5^(1/2))/2; s=r/(r-1);
a[n_]:=Floor [n*r];  (* A095280 *)
b[n_]:=Floor [n*s];  (* A095281 *)
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
A095280, A184792, A184793, A095281, A184794, A184795 *)
Select[Range[600],PrimeQ[Floor[GoldenRatio #]]&] (* Harvey P. Dale, Mar 28 2024 *)

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

cp's OEIS Frontend

A184795 Numbers m such that prime(m) is of the form floor(k*s), where s=(3+sqrt(5))/2; complement of A184793.

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A184792 Numbers k such that floor(k*r) is prime, where r = golden ratio=(1+sqrt(5))/2.

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Mathematica

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

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