A307467 The number of points, corresponding to the first n primes, and placed on the unit circle according to an algorithm using the data from A077218 (in the spirit of Ulam's spiral, and described in the COMMENTS section below), which lie on the closed arc of the unit circle from 0 to 45 degrees.

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 5, 5, 6, 7

Offset: 1

Author

Bernard X. Russo, Apr 09 2019

Keywords

• nonn
• more

Comments

Algorithm:

1. Place a dot, representing p_1, on the unit circle at (1,0), or equivalently, at angle 0 degrees, and label this point as the complex number lambda_1 = 1 = e^(i*0).

2. Place a dot, representing p_2, on the unit circle at (-1,0), or equivalently, at angle 180 degrees, and label this point as the complex number lambda_2 = -1 = e^(i*Pi).

3. For n > 2, supposing that dots representing p_1, ..., p_n have already been placed on the unit circle, place a dot, representing p_{n+1}, on the unit circle as follows: Starting at the dot representing p_n, proceed counterclockwise around the circle until you reach a dot representing one of p_1, p_2, ..., p_{n-1}.

4. Starting at this latter dot, proceed counterclockwise until you reach another dot representing one of p_1, p_2, ..., p_n.

5. Continue this process until you have traversed a(n) (of A077218) successive arcs connecting dots.

6. Proceed in a counterclockwise direction one more time to the next dot and place the dot which will represent p_{n+1} at the midpoint of the arc just traversed. Label this point as the complex number lambda_{n+1} = e^(i*theta_{n+1}).

The sequence is of interest since the points seem to cluster in the mentioned arc as well as in the closed arc from 180 degrees to 225 degrees. The sequences have been constructed by James Farrington up to n=256, which suggests the clustering. The precise clustering properties are formulated as questions in a file which is referenced below under LINKS.

Examples

			Following this procedure, we would place a dot representing p_3 on the unit circle at (0,-1), or equivalently, at angle 270 degrees, and we would place a dot representing p_4 on the unit circle at the midpoint of the arc connecting the point (-1,0) to the point (0,-1), that is, at (-1/root2,-1/root2), or equivalently, at angle 225 degrees.
Thus lambda_3 = -i = e^(3*Pi/2) and lambda_4 = e^(5*Pi/4).
		

Links

• Bernard X. Russo, Clustering of primes relative to factors of composites

Crossrefs

Uses A077218 in algorithm.