A346294 Numbers with two or more distinct prime factors such that the number and all its prime factors fall on a single straight line when they are plotted on a square spiral.

21, 24, 35, 87, 91, 99, 106, 176, 200, 273, 282, 363, 432, 507, 564, 651, 669, 951, 1333, 1445, 1805, 1837, 1963, 2669, 2813, 4163, 4557, 4625, 6321, 6643, 6685, 6723, 7225, 7567, 8333, 10152, 10252, 12826, 12877, 14761, 21409, 23317, 24651, 25337, 27391, 27419, 32039, 34225, 36673, 42029

Offset: 1

Author

Scott R. Shannon, Jul 13 2021

Keywords

• nonn

Comments

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).

Examples

			The square spiral is numbered as follows:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
21 is a term as 21 = 3 * 7, and 21 is at coordinate (-2,-2) relative to the central 1 square, 3 is at coordinate (1,1), and 7 is at coordinate (-1,-1). These three points all fall on the line y = x.
87 is a term as 87 = 3 * 29, and 87 is at coordinate (5,1), 3 is at coordinate (1,1), and 29 is at coordinate (3,1). These three points all fall on the line y = 1.
200 is a term as 200 = 2^3 * 5^2, and 200 is at coordinate (-7,4), 2 is at coordinate (1,0), and 5 is at coordinate (-1,1). These three points all fall on the line y = -x/2 + 1/2.
273 is a term as 271 = 3 * 7 * 13, and 273 is at coordinate (-8,-8), 3 is at coordinate (1,1), 7 is at coordinate (-1,-1), and 13 is at coordinate (2,2). These four points all fall on the line y = x. This is the first term with three distinct prime factors.
65793 is a term as 65793 = 3 * 7 * 13 * 241, and all these points fall on the line y = x. This is the first term with four distinct prime factors.
6118203 is a term as 6118203 = 3 * 7 * 13 * 73 * 307, and all these points fall on the line y = x. This is the first term with five distinct prime factors.
		

Links

• Scott R. Shannon, Image of the first 258 terms with their most distant divisor. Each term, shown in its correct position on the square spiral, is connected to its most distant divisor by a line, each highlighted by a square, the divisor's square being slightly smaller. The central 1 square is shown as a white square. The line colors are graduated across the spectrum to show their relative ordering. Note the higher concentration of lines along the four diagonals. Click on the image to zoom in.

Crossrefs

Cf. A027748, A214664, A214665, A330979, A335661, A335710.