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

Author

Jonathan Vos Post, Aug 09 2007

Keywords

• easy
• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A001622.

Programs

Magma

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

Mathematica

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

Python

from math import isqrt
def A130568(n): return (n+isqrt(5*n**2)&-2)|1 # Chai Wah Wu, May 22 2025

Formula

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

Extensions

More terms from Stefan Steinerberger, Aug 10 2007