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%I #26 Sep 08 2022 08:45:55
%S 2,5,7,11,19,29,31,41,43,53,59,67,73,79,83,89,97,101,103,107,113,127,
%T 131,137,149,151,173,179,181,193,197,199,223,227,229,233,239,241,251,
%U 257,263,271,277,281,311,313,337,347,349,353,359,367,373,379,383,397
%N Primes of the form floor(k*sqrt(2)).
%C 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.
%C 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.
%C 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.
%C 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?
%C [Update regarding the conjecture from Clark Kimberling, Jan 03 2011.]
%H Charles R Greathouse IV, <a href="/A184774/b184774.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) ~ sqrt(2)*n log n. - _Charles R Greathouse IV_, Apr 29 2015
%e 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,...
%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 *)
%t Select[Floor[Range[500]Sqrt[2]],PrimeQ] (* _Harvey P. Dale_, Jan 05 2019 *)
%o (PARI) is(n)=my(k=sqrtint(n^2\2)+1); sqrtint(2*k^2)==n && isprime(n) \\ _Charles R Greathouse IV_, Apr 29 2015
%o (Magma) [Floor(n*Sqrt(2)): n in [1..400] | IsPrime(Floor(n*Sqrt(2)))]; // _Vincenzo Librandi_, Apr 30 2015
%o (Python)
%o from math import isqrt
%o from itertools import count, islice
%o from sympy import isprime
%o def A184774_gen(): # generator of terms
%o return filter(isprime,(isqrt(k**2<<1) for k in count(1)))
%o A184774_list = list(islice(A184774_gen(),25)) # _Chai Wah Wu_, Jul 28 2022
%Y Cf. A001951 (Beatty sequence of sqrt(2)), A001952 (Beatty sequence of 2+sqrt(2)), A184775, A184776, A184777, A184778, A184779.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Jan 21 2011
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