A330761 Array read by antidiagonals: T(n,k) is the number of faces on a ring formed by connecting the ends of a prismatic rod whose cross-section is an n-sided regular polygon after applying a twist of k/n turns.

2, 3, 1, 4, 1, 2, 5, 1, 1, 1, 6, 1, 2, 3, 2, 7, 1, 1, 1, 1, 1, 8, 1, 2, 1, 4, 1, 2, 9, 1, 1, 3, 1, 1, 3, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2

Offset: 1

Author

Harry E. Neel, Dec 29 2019

Keywords

• nonn
• tabl

Comments

This sequence begins with two (2). A prismatic rod having two (biconvex) sides is the simplest three-dimensional construct for consideration here. Such a rod when twisted can only have one side and edge or two sides and edges. (Because the numbers of sides and edges are always equal, generally only sides or faces will be referenced.) The number of faces (surfaces) on rings generated by twisting (or not) prismatic rods is the greatest common divisor (GCD) of the number of sides of the rod, n, and the amount of twist, k, applied to the rod before forming a ring. That is that number of faces is equal to gcd(n,k). Because of the relationship of the number of sides of a prismatic rod and the variable of twist that may be applied, all prismatic rods that have a polygonal cross-section that is prime in number (2, 3, 5, 7, 11, etc.) and formed into rings will always have only 1 face (surface) if twisted by any amount that is not a multiple of the prime. E.g., a prismatic rod with a pentagonal cross-section will have 5 faces if left untwisted or if twisted 5/5, 10/5, etc. Any other amount of rotation will always produce 1 face.

The direction of rotation does not matter. This is an infinite sequence in the sense that primes are infinite (not to mention composites). However, it is noted that actual constructs have limitations.

Examples

			A prismatic rod having a cross-section that is an octagon will have:
8 faces if no twist is applied or if the amount of twisting is a multiple of 8: 0/8, 8/8, 16/8, etc.;
4 faces if the amount of twisting is 4/8, 12/8, etc.
2 faces if the amount of twisting is 2/8, 6/8, 10/8, etc.
1 face if the amount of twisting is 1/8, 3/8, 5/8, 7/8, 9/8, etc.
Note that the number of faces is equal to gcd(n,k) where n=number of sides of the prismatic rod and k=amount of twist applied to the rod.
T(n,k) as a table begins:
(n=number of sides of polygon; k=amount of fractional twist applied)
   n |k=0  1  2  3  4  5  6  7  8  9 10 11 12 13 ...
  ---+-------------------------------------------
   2 |  2  1  2  1  2  1  2  1  2  1  2  1  2  1...
   3 |  3  1  1  3  1  1  3  1  1  3  1  1  3  1...
   4 |  4  1  2  1  4  1  2  1  4  1  2  1  4  1...
   5 |  5  1  1  1  1  5  1  1  1  1  5  1  1  1...
   6 |  6  1  2  3  2  1  6  1  2  3  2  1  6  1...
   7 |  7  1  1  1  1  1  1  7  1  1  1  1  1  1...
   8 |  8  1  2  1  4  1  2  1  8  1  2  1  4  1...
   9 |  9  1  1  3  1  1  3  1  1  9  1  1  3  1...
  10 | 10  1  2  1  2  5  2  1  2  1 10  1  2  1...
  11 | 11  1  1  1  1  1  1  1  1  1  1 11  1  1...
  12 | 12  1  2  3  4  1  6  1  4  3  2  1 12  1...
  13 | 13  1  1  1  1  1  1  1  1  1  1  1  1 13...
  ...
		

Links

• Martin Gardner, Mathematical Games, A Möbius band has a finite thickness, and so it is actually a twisted prism, Scientific American, August, 1978, 18-25.

Crossrefs

Subtable of A109004.

Formula

T(n, k) = gcd(n, k).