A132222 Beatty sequence 1+2*floor(n*Pi/2), which contains infinitely many primes.

1, 3, 7, 9, 13, 15, 19, 21, 25, 29, 31, 35, 37, 41, 43, 47, 51, 53, 57, 59, 63, 65, 69, 73, 75, 79, 81, 85, 87, 91, 95, 97, 101, 103, 107, 109, 113, 117, 119, 123, 125, 129, 131, 135, 139, 141, 145, 147, 151, 153, 157, 161, 163, 167, 169, 173, 175, 179, 183, 185, 189

Offset: 0

Author

Jonathan Vos Post, Aug 14 2007

Keywords

• easy
• nonn

Comments

The primes in this entirely odd sequence begin 3, 7, 13, 19, 29, 31, 37, 41, 43, 47, 53, 59, 73, 79, 97, 101. By the theorems in Banks, there are an infinite number of primes in this sequence.

Examples

			a(0) = 1 because 1 + 2*floor(0*Pi) = 1 + 2*0 = 1 + 0 = 1.
a(1) = 3 because 1 + 2*floor(1*Pi/2) = 1 + 2*floor(1.5707963) = 1 + 2*1 = 3.
a(2) = 7 because 1 + 2*floor(2*Pi/2) = 1 + 2*floor(3.1415926) = 1 + 2*3 = 7.
a(3) = 9 because 1 + 2*floor(3*Pi/2) = 1 + 2*floor(4.7123889) = 1 + 2*4 = 9.
a(4) = 13 because 1 + 2*floor(4*Pi/2) = 1 + 2*floor(6.2831853) = 1 + 2*6 = 13.
a(5) = 15 because 1 + 2*floor(5*Pi/2) = 1 + 2*floor(7.8539816) = 1 + 2*7 = 15.
a(7) = 21 because 1 + 2*floor(7*Pi/2) = 1 + 2*floor(10.995574) = 1 + 2*10 = 21.
		

Links

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

Crossrefs

Cf. A019669, A130568, A140758.

Programs

Mathematica

Table[1 + 2*Floor[n*Pi/2], {n, 0, 60}] (* Stefan Steinerberger, Sep 02 2007 *)

Formula

a(n) = 1 + 2*floor(n*Pi/2).

a(n) = 1 + 2*A140758(n). - L. Edson Jeffery, Mar 16 2013

Extensions

More terms from Stefan Steinerberger, Sep 02 2007