A055684 - OEIS

%I #41 Nov 20 2020 17:14:18

%S 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,

%T 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,

%U 29,14,17,15,23,9,32,15,21,11,34,11,35,17,19,17,29,11

%N Number of different n-pointed stars.

%C Does not count rotations or reflections.

%C 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

%C 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

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

%H Lei Zhou, <a href="/A055684/b055684.txt">Table of n, a(n) for n = 3..10002</a>

%H Alexander Bogomolny, <a href="http://www.cut-the-knot.org/Generalization/PolyStar.shtml">Polygons: formality and intuition.</a>. Includes applet to draw star polygons.

%H Vi Hart, <a href="http://youtu.be/CfJzrmS9UfY">Doodling in Math Class: Stars</a>, Video (2010).

%H Hugo Pfoertner, <a href="/A055684/a055684.pdf">Star-shaped regular polygons up to n=25.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StarPolygon.html">Star Polygon</a>

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

%F a(n) + A082023(n) = A140106(n). - _Gus Wiseman_, Oct 28 2020

%e The first star has five points and is unique. The next is the seven pointed star and it comes in two varieties.

%e From _Gus Wiseman_, Oct 28 2020: (Start)

%e The a(5) = 1 through a(17) = 7 irreducible pairs > 1 (shown as fractions, empty column indicated by dot):

%e   2/3  .  2/5  3/5  2/7  3/7  2/9  5/7  2/11  3/11  2/13  3/13  2/15

%e           3/4       4/5       3/8       3/10  5/9   4/11  5/11  3/14

%e                               4/7       4/9         7/8   7/9   4/13

%e                               5/6       5/8                     5/12

%e                                         6/7                     6/11

%e                                                                 7/10

%e                                                                 8/9

%e (End)

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

%t Table[(EulerPhi[n]-2)/2, {n, 3, 50}]

%t Table[Length[Select[IntegerPartitions[n,{2}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}] (* _Gus Wiseman_, Oct 28 2020 *)

%Y Cf. A023022.

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

%Y A055684*2 is the ordered version.

%Y A082023 counts the complement (reducible pairs > 1).

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

%Y A000837 counts relatively prime partitions, with strict case A078374.

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

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

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

%Y A327516 counts pairwise coprime partitions, with strict case A305713.

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

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

%K nonn,easy

%O 3,5

%A _Robert G. Wilson v_, Jun 09 2000