A317626 Intersections with the x-axis of a bouncing ball on a Sophie Germain billiard table.

2, 4, 8, 10, 14, 18, 28, 30, 38, 44, 58, 60, 64, 78, 80, 84, 94, 98, 120, 140, 144, 148, 164, 170, 198, 214, 218, 220, 228, 240, 248, 254, 270, 304, 318, 338, 340, 344, 350, 368, 408, 410, 430, 470, 480, 484, 494, 500, 504, 520, 528, 534, 578, 604, 630, 634, 644, 658

Offset: 1

Author

Hilko Koning, Aug 02 2018

Keywords

• nonn

Comments

In the first quadrant of a coordinate system define a rectangular Sophie Germain billiard table with width p and length 2p+1, with vertices (0,0), (p,0), (p,2p+1) and (0,2p+1). A billiard ball (considered to be a point) starts from (0,0) at an angle of 45 degrees and hits the sides exactly p times until it hits the x-axis. The sequence gives the intersections with the x-axis of consecutive Sophie Germain prime numbers (p > 3) after p bounces.

The sum of all crossed lattice points (including the rectangle sides) is the sum of crossed points left under, right middle and left up respectively ((p+7)/6)^2 + (p+1)(p+4)/18 + (p+1)(p+7)/36 = ((p+4)/3)^2 (see bouncing examples).

The enclosed areas in the Sophie Germain billiard table also correspond to ((p+4)/3)^2.

The number of trajectories is a subsequence of A176045.

The number of trajectories with slope +1 or with slope -1 is a subsequence of A124485.

The sum of a term of this sequence and the corresponding Sophie Germain prime is A317510 and it appears that this is a subsequence of A179882. Checked up to and including 33295 of A317510 (Sophie Germain prime 24971).

Links

• Samuel King, Billiard Simulator
• Hilko Koning, Some bouncing examples.

Crossrefs

Cf. A005384, A151922, A176045, A124485, A179882, A317510.

Programs

GAP

a:=[];; for p in [3..2000] do if IsPrime(p) and IsPrime(2*p+1) then Add(a,(p+1)/3); fi; od; a; # Muniru A Asiru, Aug 28 2018

Mathematica

lst = {}; Do[If[PrimeQ[p] && PrimeQ[2 p + 1], AppendTo[lst, (p + 1)/3]], {p, 5, 2*10^3}]; lst
(Select[Prime@ Range[3, 300], PrimeQ[2# + 1] &] + 1)/3 (* Robert G. Wilson v, Aug 02 2018 *)

PARI

lista(nn) = forprime(p=2, nn, if (isprime(2*p+1), print1((p+1)/3, ", "));); \\ Michel Marcus, Aug 25 2018

Formula

a(n) = (A005384(n)+1)/3 for n>=3. - Michel Marcus, Aug 25 2018