A184774 Primes of the form floor(k*sqrt(2)).

2, 5, 7, 11, 19, 29, 31, 41, 43, 53, 59, 67, 73, 79, 83, 89, 97, 101, 103, 107, 113, 127, 131, 137, 149, 151, 173, 179, 181, 193, 197, 199, 223, 227, 229, 233, 239, 241, 251, 257, 263, 271, 277, 281, 311, 313, 337, 347, 349, 353, 359, 367, 373, 379, 383, 397

Offset

1,1

Comments

Let N={1,2,...}, L={floor(n*sqrt(2)): n in N} and U={2n+L(n): n in N}. Every prime is in L or U, since the union of the (disjoint) sets L and U is N.

The conjecture formerly posted here, that "if r is an irrational number and 1<r<2, then there are infinitely many primes in the set L={floor(n*r)}," is proved in I. M. Vinogradov, The Method of Trigonometrical Sums in the Theory of Numbers, (1954), page 180.

Note that every prime not in L is in U={floor(n*s}, where s=r/(r-1). That is, Beatty sequences partition the primes into two infinite classes.

The conjecture generalized: if r is a positive irrational number and h is a real number, then each of the sets {floor(n*r+h)}, {round(n*r+h)}, and {ceiling(n*r+h)} contains infinitely many primes. Can the method in Vinogradov be extended to cover these cases?

[Update regarding the conjecture from Clark Kimberling, Jan 03 2011.]

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) ~ sqrt(2)*n log n. - Charles R Greathouse IV, Apr 29 2015

Example

The sequence L(n)=floor(n*sqrt(2)) begins with 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19,..., which includes primes L(2)=2, L(4)=5, L(5)=7,...

Mathematica

r=2^(1/2); s=r/(r-1);
a[n_]:=Floor [n*r];  (* A001951 *)
b[n_]:=Floor [n*s];  (* A001952 *)
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
A184774, A184775, A184776 , A184777, A184778, A184779 *)
Select[Floor[Range[500]Sqrt[2]], PrimeQ] (* Harvey P. Dale, Jan 05 2019 *)

Prog

(PARI) is(n)=my(k=sqrtint(n^2\2)+1); sqrtint(2*k^2)==n && isprime(n) \\ Charles R Greathouse IV, Apr 29 2015
(Magma) [Floor(n*Sqrt(2)): n in [1..400] | IsPrime(Floor(n*Sqrt(2)))]; // Vincenzo Librandi, Apr 30 2015
(Python)
from math import isqrt
from itertools import count, islice
from sympy import isprime
def A184774_gen(): # generator of terms
    return filter(isprime, (isqrt(k**2<<1) for k in count(1)))
A184774_list = list(islice(A184774_gen(), 25)) # Chai Wah Wu, Jul 28 2022

Crossrefs

Cf. A001951 (Beatty sequence of sqrt(2)), A001952 (Beatty sequence of 2+sqrt(2)), A184775, A184776, A184777, A184778, A184779.

Sequence in context: A116635 A340759 A252280 * A038884 A338345 A252282

Adjacent sequences: A184771 A184772 A184773 * A184775 A184776 A184777

Keyword

nonn,easy

Author

Clark Kimberling, Jan 21 2011

Status

approved