A184775 Numbers k such that floor(k*sqrt(2)) is prime.

2, 4, 5, 8, 14, 21, 22, 29, 31, 38, 42, 48, 52, 56, 59, 63, 69, 72, 73, 76, 80, 90, 93, 97, 106, 107, 123, 127, 128, 137, 140, 141, 158, 161, 162, 165, 169, 171, 178, 182, 186, 192, 196, 199, 220, 222, 239, 246, 247, 250, 254, 260, 264, 268, 271, 281, 284, 298, 305, 311, 318

Offset: 1

Author

Clark Kimberling, Jan 21 2011

Keywords

• nonn
• easy

Comments

Chua, Park, & Smith prove a general result that implies that, for any m, there is a constant C(m) such that a(n+m) - a(n) < C(m) infinitely often. - Charles R Greathouse IV, Jul 01 2022

Examples

			See A184774.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Lynn Chua, Soohyun Park, and Geoffrey D. Smith, Bounded gaps between primes in special sequences, Proceedings of the AMS, Volume 143, Number 11 (November 2015), pp. 4597-4611. arXiv:1407.1747 [math.NT]

Crossrefs

Cf. A001951, A184774, A184776.

Programs

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 *)

PARI

isok(n) = isprime(floor(n*sqrt(2))); \\ Michel Marcus, Apr 10 2018

PARI

is(n)=isprime(sqrtint(2*n^2)) \\ Charles R Greathouse IV, Jul 01 2022

Python

from itertools import count, islice
from math import isqrt
from sympy import isprime
def A184775_gen(): # generator of terms
    return filter(lambda k:isprime(isqrt(k**2<<1)), count(1))
A184775_list = list(islice(A184775_gen(),25)) # Chai Wah Wu, Jul 28 2022