This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A130568 #32 May 23 2025 03:53:43
%S A130568 1,3,7,9,13,17,19,23,25,29,33,35,39,43,45,49,51,55,59,61,65,67,71,75,
%T A130568 77,81,85,87,91,93,97,101,103,107,111,113,117,119,123,127,129,133,135,
%U A130568 139,143,145,149,153,155,159,161,165,169,171,175,177,181,185,187,191,195
%N A130568 Generalized Beatty sequence 1+2*floor(n*phi), which contains infinitely many primes.
%C A130568 The primes in this entirely odd sequence begin 3, 7, 13, 17, 19, 23, 29. By the theorems in Banks, there are an infinite number of primes in this sequence.
%C A130568 Conjecture: Sequence gives n of A163873 whose connection to a(n) crosses (in the tree of A163873) another path. Is this generalizable in any way for A163874, A163875? - Daniel Platt (d.platt(AT)web.de), Sep 14 2009
%H A130568 William D. Banks and Igor E. Shparlinski, <a href="http://arXiv.org/abs/0708.1015">Prime numbers with Beatty sequences</a>, arXiv:0708.1015 [math.NT], 7 Aug 2007.
%F A130568 a(n) = 1+2*floor(n*phi), where phi = (1 + sqrt(5))/2.
%e A130568 a(0) = 1 + 2*floor(0*phi) = 1 + 2*0 = 1.
%e A130568 a(1) = 1 + 2*floor(1*phi) = 1 + 2*floor(1.6180) = 1 + 2*1 = 3.
%e A130568 a(2) = 1 + 2*floor(2*phi) = 1 + 2*floor(3.2360) = 1 + 2*3 = 7.
%e A130568 a(3) = 1 + 2*floor(3*phi) = 1 + 2*floor(4.8541) = 1 + 2*4 = 9.
%e A130568 a(4) = 1 + 2*floor(4*phi) = 1 + 2*floor(6.4721) = 1 + 2*6 = 13.
%e A130568 a(5) = 1 + 2*floor(5*phi) = 1 + 2*floor(8.0901) = 1 + 2*8 = 17.
%e A130568 a(6) = 1 + 2*floor(6*phi) = 1 + 2*floor(9.7082) = 1 + 2*9 = 19.
%e A130568 a(7) = 1 + 2*floor(7*phi) = 1 + 2*floor(11.3262) = 1 + 2*11 = 23.
%e A130568 a(8) = 1 + 2*floor(8*phi) = 1 + 2*floor(12.9442) = 1 + 2*12 = 25.
%e A130568 a(9) = 1 + 2*floor(9*phi) = 1 + 2*floor(14.5623) = 1 + 2*14 = 29.
%e A130568 a(10) = 1 + 2*floor(10*phi) = 1 + 2*floor(16.1803) = 1 + 2*16 = 33.
%t A130568 Table[1 + 2*Floor[n*(Sqrt[5] + 1)/2], {n, 0, 80}] (* _Stefan Steinerberger_, Aug 10 2007 *)
%o A130568 (Magma) [1+2*Floor(n*((1+Sqrt(5))/2)): n in [0..60]]; // _Vincenzo Librandi_, Sep 17 2015
%o A130568 (Python)
%o A130568 from math import isqrt
%o A130568 def A130568(n): return (n+isqrt(5*n**2)&-2)|1 # _Chai Wah Wu_, May 22 2025
%Y A130568 Cf. A001622.
%K A130568 easy,nonn
%O A130568 0,2
%A A130568 _Jonathan Vos Post_, Aug 09 2007
%E A130568 More terms from _Stefan Steinerberger_, Aug 10 2007