A055684 Number of different n-pointed stars.

0, 0, 1, 0, 2, 1, 2, 1, 4, 1, 5, 2, 3, 3, 7, 2, 8, 3, 5, 4, 10, 3, 9, 5, 8, 5, 13, 3, 14, 7, 9, 7, 11, 5, 17, 8, 11, 7, 19, 5, 20, 9, 11, 10, 22, 7, 20, 9, 15, 11, 25, 8, 19, 11, 17, 13, 28, 7, 29, 14, 17, 15, 23, 9, 32, 15, 21, 11, 34, 11, 35, 17, 19, 17, 29, 11

Offset: 3

Author

Robert G. Wilson v, Jun 09 2000

Keywords

• nonn
• easy

Comments

Does not count rotations or reflections.

This is also the distinct ways of writing a number as the sum of two positive integers greater than one that are coprimes. - Lei Zhou, Mar 19 2014

Equivalently, a(n) is the number of relatively prime 2-part partitions of n without 1's. The Heinz numbers of these partitions are the intersection of A001358 (pairs), A005408 (no 1's), and A000837 (relatively prime) or A302696 (pairwise coprime). - Gus Wiseman, Oct 28 2020

Examples

			The first star has five points and is unique. The next is the seven pointed star and it comes in two varieties.
From _Gus Wiseman_, Oct 28 2020: (Start)
The a(5) = 1 through a(17) = 7 irreducible pairs > 1 (shown as fractions, empty column indicated by dot):
  2/3  .  2/5  3/5  2/7  3/7  2/9  5/7  2/11  3/11  2/13  3/13  2/15
          3/4       4/5       3/8       3/10  5/9   4/11  5/11  3/14
                              4/7       4/9         7/8   7/9   4/13
                              5/6       5/8                     5/12
                                        6/7                     6/11
                                                                7/10
                                                                8/9
(End)
		

References

• Mark A. Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, New York, 1999, page 58.

Links

• Lei Zhou, Table of n, a(n) for n = 3..10002
• Alexander Bogomolny, Polygons: formality and intuition.. Includes applet to draw star polygons.
• Vi Hart, Doodling in Math Class: Stars, Video (2010).
• Hugo Pfoertner, Star-shaped regular polygons up to n=25.
• Eric Weisstein's World of Mathematics, Star Polygon

Crossrefs

Cf. A023022.

Cf. A053669 smallest skip increment, A102302 skip increment of densest star polygon.

A055684*2 is the ordered version.

A082023 counts the complement (reducible pairs > 1).

A220377, A337563, and A338332 count triples instead of pairs.

A000837 counts relatively prime partitions, with strict case A078374.

A002865 counts partitions with no 1's, with strict case A025147.

A007359 and A337485 count pairwise coprime partitions with no 1's.

A302698 counts relatively prime partitions with no 1's, with strict case A337452.

A327516 counts pairwise coprime partitions, with strict case A305713.

A337450 counts relatively prime compositions with no 1's, with strict case A337451.

Cf. A001399, A101268, A140106, A337461, A337462, A338333.

Programs

Maple

with(numtheory): A055684 := n->(phi(n)-2)/2; seq(A055684(n), n=3..100);

Mathematica

Table[(EulerPhi[n]-2)/2, {n, 3, 50}]
Table[Length[Select[IntegerPartitions[n,{2}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 28 2020 *)

Formula

a(n) = A023022(n) - 1.

a(n) + A082023(n) = A140106(n). - Gus Wiseman, Oct 28 2020