A130568 Generalized Beatty sequence 1+2*floor(n*phi), which contains infinitely many primes.

1, 3, 7, 9, 13, 17, 19, 23, 25, 29, 33, 35, 39, 43, 45, 49, 51, 55, 59, 61, 65, 67, 71, 75, 77, 81, 85, 87, 91, 93, 97, 101, 103, 107, 111, 113, 117, 119, 123, 127, 129, 133, 135, 139, 143, 145, 149, 153, 155, 159, 161, 165, 169, 171, 175, 177, 181, 185, 187, 191, 195

Offset

0,2

Comments

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.

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

Links

Table of n, a(n) for n=0..60.

William D. Banks, Igor E. Shparlinski, Prime numbers with Beatty sequences, arXiv:0708.1015 [math.NT], 7 Aug 2007.

Formula

a(n) = 1+2*floor(n*phi), where phi = (1 + sqrt(5))/2.

Example

a(0) = 1 + 2*floor(0*phi) = 1 + 2*0 = 1.
a(1) = 1 + 2*floor(1*phi) = 1 + 2*floor(1.6180) = 1 + 2*1 = 3.
a(2) = 1 + 2*floor(2*phi) = 1 + 2*floor(3.2360) = 1 + 2*3 = 7.
a(3) = 1 + 2*floor(3*phi) = 1 + 2*floor(4.8541) = 1 + 2*4 = 9.
a(4) = 1 + 2*floor(4*phi) = 1 + 2*floor(6.4721) = 1 + 2*6 = 13.
a(5) = 1 + 2*floor(5*phi) = 1 + 2*floor(8.0901) = 1 + 2*8 = 17.
a(6) = 1 + 2*floor(6*phi) = 1 + 2*floor(9.7082) = 1 + 2*9 = 19.
a(7) = 1 + 2*floor(7*phi) = 1 + 2*floor(11.3262) = 1 + 2*11 = 23.
a(8) = 1 + 2*floor(8*phi) = 1 + 2*floor(12.9442) = 1 + 2*12 = 25.
a(9) = 1 + 2*floor(9*phi) = 1 + 2*floor(14.5623) = 1 + 2*14 = 29.
a(10) = 1 + 2*floor(10*phi) = 1 + 2*floor(16.1803) = 1 + 2*16 = 33.

Mathematica

Table[1 + 2*Floor[n*(Sqrt[5] + 1)/2], {n, 0, 80}] (* Stefan Steinerberger, Aug 10 2007 *)

Prog

(Magma) [1+2*Floor(n*((1+Sqrt(5))/2)): n in [0..60]]; // Vincenzo Librandi, Sep 17 2015

Crossrefs

Cf. A001622.

Sequence in context: A063204 A236208 A284884 * A143803 A284894 A020497

Adjacent sequences: A130565 A130566 A130567 * A130569 A130570 A130571

Keyword

easy,nonn

Author

Jonathan Vos Post, Aug 09 2007

Extensions

More terms from Stefan Steinerberger, Aug 10 2007

Status

approved