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%I #26 Sep 08 2022 08:45:30
%S 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 77,81,85,87,91,93,97,101,103,107,111,113,117,119,123,127,129,133,135,
%U 139,143,145,149,153,155,159,161,165,169,171,175,177,181,185,187,191,195
%N Generalized Beatty sequence 1+2*floor(n*phi), which contains infinitely many primes.
%C 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 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 William D. Banks, 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 a(n) = 1+2*floor(n*phi), where phi = (1 + sqrt(5))/2.
%e a(0) = 1 + 2*floor(0*phi) = 1 + 2*0 = 1.
%e a(1) = 1 + 2*floor(1*phi) = 1 + 2*floor(1.6180) = 1 + 2*1 = 3.
%e a(2) = 1 + 2*floor(2*phi) = 1 + 2*floor(3.2360) = 1 + 2*3 = 7.
%e a(3) = 1 + 2*floor(3*phi) = 1 + 2*floor(4.8541) = 1 + 2*4 = 9.
%e a(4) = 1 + 2*floor(4*phi) = 1 + 2*floor(6.4721) = 1 + 2*6 = 13.
%e a(5) = 1 + 2*floor(5*phi) = 1 + 2*floor(8.0901) = 1 + 2*8 = 17.
%e a(6) = 1 + 2*floor(6*phi) = 1 + 2*floor(9.7082) = 1 + 2*9 = 19.
%e a(7) = 1 + 2*floor(7*phi) = 1 + 2*floor(11.3262) = 1 + 2*11 = 23.
%e a(8) = 1 + 2*floor(8*phi) = 1 + 2*floor(12.9442) = 1 + 2*12 = 25.
%e a(9) = 1 + 2*floor(9*phi) = 1 + 2*floor(14.5623) = 1 + 2*14 = 29.
%e a(10) = 1 + 2*floor(10*phi) = 1 + 2*floor(16.1803) = 1 + 2*16 = 33.
%t Table[1 + 2*Floor[n*(Sqrt[5] + 1)/2], {n, 0, 80}] (* _Stefan Steinerberger_, Aug 10 2007 *)
%o (Magma) [1+2*Floor(n*((1+Sqrt(5))/2)): n in [0..60]]; // _Vincenzo Librandi_, Sep 17 2015
%Y Cf. A001622.
%K easy,nonn
%O 0,2
%A _Jonathan Vos Post_, Aug 09 2007
%E More terms from _Stefan Steinerberger_, Aug 10 2007
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