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A184777 Primes of the form 2k + floor(k*sqrt(2)).

%I A184777 #14 Mar 06 2025 11:53:13
%S A184777 3,13,17,23,37,47,61,71,109,139,157,163,167,191,211,269,283,293,307,
%T A184777 317,331,389,409,419,433,443,457,467,491,563,577,587,607,617,631,641,
%U A184777 727,751,757,761,809,829,839,853,863,877,887,911,983,1031,1051,1061,1109,1123,1171,1181,1201,1229,1249,1259,1283,1297,1307,1321,1399,1423,1427,1433,1447,1451,1471,1481,1543,1553,1567,1597,1601,1621,1669,1693,1741,1789,1823,1847,1867,1877,1901
%N A184777 Primes of the form 2k + floor(k*sqrt(2)).
%H A184777 G. C. Greubel, <a href="/A184777/b184777.txt">Table of n, a(n) for n = 1..10000</a>
%e A184777 See A184774.
%t A184777 r=2^(1/2); s=r/(r-1);
%t A184777 a[n_]:=Floor [n*r];  (* A001951 *)
%t A184777 b[n_]:=Floor [n*s];  (* A001952 *)
%t A184777 Table[a[n],{n,1,120}]
%t A184777 t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]], {n,1,600}]; t1
%t A184777 t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2,n]], {n,1,600}]; t2
%t A184777 t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3,n]],{n,1,300}]; t3
%t A184777 t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}]; t4
%t A184777 t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}]; t5
%t A184777 t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6,n]],{n,1,300}]; t6
%t A184777 (* the lists t1,t2,t3,t4,t5,t6 match the sequences
%t A184777 A184774, A184775, A184776 ,A184777, A184778, A184779 *)
%t A184777 Select[Table[2k+Floor[k Sqrt[2]],{k,1000}],PrimeQ] (* _Harvey P. Dale_, Mar 06 2025 *)
%o A184777 (Python)
%o A184777 from math import isqrt
%o A184777 from itertools import count, islice
%o A184777 from sympy import isprime
%o A184777 def A184777_gen(): # generator of terms
%o A184777     return filter(isprime,((k<<1)+isqrt(k**2<<1) for k in count(1)))
%o A184777 A184777_list = list(islice(A184777_gen(),25)) # _Chai Wah Wu_, Jul 28 2022
%Y A184777 Cf. A184774, A184778, A184779.
%K A184777 nonn
%O A184777 1,1
%A A184777 _Clark Kimberling_, Jan 21 2011