A184775 - OEIS

%I #25 Jul 29 2022 09:51:13

%S 2,4,5,8,14,21,22,29,31,38,42,48,52,56,59,63,69,72,73,76,80,90,93,97,

%T 106,107,123,127,128,137,140,141,158,161,162,165,169,171,178,182,186,

%U 192,196,199,220,222,239,246,247,250,254,260,264,268,271,281,284,298,305,311,318

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

%C 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

%H G. C. Greubel, <a href="/A184775/b184775.txt">Table of n, a(n) for n = 1..10000</a>

%H Lynn Chua, Soohyun Park, and Geoffrey D. Smith, <a href="https://arxiv.org/abs/1407.1747">Bounded gaps between primes in special sequences</a>, Proceedings of the AMS, Volume 143, Number 11 (November 2015), pp. 4597-4611. arXiv:1407.1747 [math.NT]

%e See A184774.

%t r=2^(1/2); s=r/(r-1);

%t a[n_]:=Floor [n*r];  (* A001951 *)

%t b[n_]:=Floor [n*s];  (* A001952 *)

%t Table[a[n],{n,1,120}]

%t t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]], {n,1,600}]; t1

%t t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2,n]], {n,1,600}]; t2

%t t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3,n]],{n,1,300}]; t3

%t t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}]; t4

%t t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}]; t5

%t t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6,n]],{n,1,300}]; t6

%t (* the lists t1,t2,t3,t4,t5,t6 match the sequences

%t A184774, A184775, A184776 ,A184777, A184778, A184779 *)

%o (PARI) isok(n) = isprime(floor(n*sqrt(2))); \\ _Michel Marcus_, Apr 10 2018

%o (PARI) is(n)=isprime(sqrtint(2*n^2)) \\ _Charles R Greathouse IV_, Jul 01 2022

%o (Python)

%o from itertools import count, islice

%o from math import isqrt

%o from sympy import isprime

%o def A184775_gen(): # generator of terms

%o     return filter(lambda k:isprime(isqrt(k**2<<1)), count(1))

%o A184775_list = list(islice(A184775_gen(),25)) # _Chai Wah Wu_, Jul 28 2022

%Y Cf. A001951, A184774, A184776.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jan 21 2011