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%I A184775 #25 Jul 29 2022 09:51:13
%S A184775 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 A184775 106,107,123,127,128,137,140,141,158,161,162,165,169,171,178,182,186,
%U A184775 192,196,199,220,222,239,246,247,250,254,260,264,268,271,281,284,298,305,311,318
%N A184775 Numbers k such that floor(k*sqrt(2)) is prime.
%C A184775 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 A184775 G. C. Greubel, <a href="/A184775/b184775.txt">Table of n, a(n) for n = 1..10000</a>
%H A184775 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 A184775 See A184774.
%t A184775 r=2^(1/2); s=r/(r-1);
%t A184775 a[n_]:=Floor [n*r];  (* A001951 *)
%t A184775 b[n_]:=Floor [n*s];  (* A001952 *)
%t A184775 Table[a[n],{n,1,120}]
%t A184775 t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]], {n,1,600}]; t1
%t A184775 t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2,n]], {n,1,600}]; t2
%t A184775 t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3,n]],{n,1,300}]; t3
%t A184775 t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}]; t4
%t A184775 t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}]; t5
%t A184775 t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6,n]],{n,1,300}]; t6
%t A184775 (* the lists t1,t2,t3,t4,t5,t6 match the sequences
%t A184775 A184774, A184775, A184776 ,A184777, A184778, A184779 *)
%o A184775 (PARI) isok(n) = isprime(floor(n*sqrt(2))); \\ _Michel Marcus_, Apr 10 2018
%o A184775 (PARI) is(n)=isprime(sqrtint(2*n^2)) \\ _Charles R Greathouse IV_, Jul 01 2022
%o A184775 (Python)
%o A184775 from itertools import count, islice
%o A184775 from math import isqrt
%o A184775 from sympy import isprime
%o A184775 def A184775_gen(): # generator of terms
%o A184775     return filter(lambda k:isprime(isqrt(k**2<<1)), count(1))
%o A184775 A184775_list = list(islice(A184775_gen(),25)) # _Chai Wah Wu_, Jul 28 2022
%Y A184775 Cf. A001951, A184774, A184776.
%K A184775 nonn,easy
%O A184775 1,1
%A A184775 _Clark Kimberling_, Jan 21 2011

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