cp's OEIS Frontend

On a spiral spiral plot the position of a number along with all its prime factors, where the number has at least two distinct prime factors. The sequence lists those numbers for which all these points can be connected by a single straight line.

The first term with two prime factors is 21, the first with three is 273, the first with four is 65793, and the first with five is 6118203. Almost all of the later numbers lie on lines with gradient +-1 passing through or very close to the central 1 square. In general there is a concentration of term on these diagonals; see the linked image.

There are 258 terms for numbers below 100 million. In that range the largest prime factor to appear is for 69672413 = 29 * 2402497, where 2402497 has coordinate (-771,775) relative to the central 1 square, 29 is at coordinate (3,1), while the term 69672413 is at coordinate (4174,-4170).

A number is not visible from the current number if, given it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is greater than 1.

See A331400 for the points visible from the starting 1 number.

On the standard square spiral a number is not visible from the current number if, given it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is greater than 1. For this sequence at least one other number must also exist on the line connecting these two numbers for them to be hidden from each other. Most visited primes are stepped over by subsequent terms. See the first linked image.

See A331400 for the points visible from the starting 1 number.

A number is not visible from the current number if, given it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is greater than 1.

As n increases the vast majority of primes are on the same square ring of numbers as the current prime. However occasionally, especially for primes inside the right side quadrant, the next prime is on an outer or inner ring which causes the step to make a diagonal line. See the linked images. The largest diagonal step after 50000 terms is one at step 43936 between primes 532981 and 531457 which is seen as the long violet diagonal line from the top-left to the bottom-right in the image for these terms. No other such diagonal line is seen up to 10^6 terms.

The sequences lists the areas of the triangles formed by joining three consecutive primes, A000040(n), A000040(n+1), and A000040(n+2), as vertices on the Ulam spiral. As n increases the majority of terms are zero as most of the consecutive primes triples will fall on the same vertical or horizontal line forming the square spiral; only those primes near the corners of the spiral will form nonzero area triangles.

Assuming the truth of the Legendre conjecture one can show all areas will be integer values. Consider that the area, A, of a triangle is given by half the magnitude of the cross product of the vectors from the second prime of the triple to the first and third primes, i.e., A = |x_1*y_2 - y_1*x_2|/2. Any two primes on the same vertical or horizontal line of the spiral will always be a multiple of two units apart, so either x_1 = 2*k, y_1 = 0, or x_1 = 0, y_1 = 2*k where k is an integer with |k| >= 0. Assuming the third prime is not on the same line then A will be an even number divided by 2, which is always an integer. The only possibility for A being a non-integer is for all three primes to lie on three different vertical and/or horizontal spiral lines. Note that only the lower-right corner of the spiral has an even number. Therefore if we start on any right vertical line moving counterclockwise one complete revolution all primes will be an even number from the first three visited corners, so any vector connecting these primes will be of the form (2*j,2*k), implying once again the resulting triangle will have an integer value. So the remaining possibility is that the path between the three consecutive primes crosses the south-east corner at least once, for example the first prime is on the lower horizontal line and then the second is on the adjacent vertical right line. Such an example would be 23 to 29. But now, due to the above restriction that the next prime cannot be on the top, left, or bottom line if the proceeding prime is on the right vertical line, the third prime would need to form a path of one complete revolution and be on the next outer right vertical line. In the example case given this means it would have to be 51 or more. But in completing this revolution the path crosses both the top-left corner of the spiral, which is next to the numbers of the form (2*p)^2, and also the bottom right corner, which is next to numbers of the form (2*p+1)^2, and so it crosses consecutive squares without forming a prime. This violates the Legendre conjecture which, if true, therefore implies all triangles between three consecutive primes on the Ulam spiral will have an integer area.

For an Ulam spiral of size 20001 by 20001, with largest prime just over 400 million, the largest triangle area is 5160, between consecutive primes 364008101, 364008181 and 364008371. The first occurrence of three consecutive triangles with the same area, with area > 0, is for primes (2293,2297,2309), (2297,2309,2311), (2347,2351,2357), all of which form a triangle of area 8. Sixteen other runs with three consecutive triangles with the same area were also found, but no run of four triangles. The smallest triangle area which has not been formed is 79, although this minimum value slowly increases as the spiral gets larger, so it is likely, but unknown, that eventually triangles of all integer values are created.

The initial alive cells are at coordinates x=A214664(i), y=A214665(i) for i=1..n.

For the first 7 terms of this sequence we have a(n)=n since those initial configurations do not lead to complex enough patterns that increase the number of alive cells beyond the initial number of alive cells.

The definition includes the possibility that a glider gun (or a similar pattern) is created which will result in an unbounded number of alive cells.