cp's OEIS Frontend

A055684 Number of different n-pointed stars.

%I A055684 #42 Feb 16 2025 08:32:43
%S A055684 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 A055684 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 A055684 29,14,17,15,23,9,32,15,21,11,34,11,35,17,19,17,29,11
%N A055684 Number of different n-pointed stars.
%C A055684 Does not count rotations or reflections.
%C A055684 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 A055684 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 A055684 Mark A. Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, New York, 1999, page 58.
%H A055684 Lei Zhou, <a href="/A055684/b055684.txt">Table of n, a(n) for n = 3..10002</a>
%H A055684 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 A055684 Vi Hart, <a href="http://youtu.be/CfJzrmS9UfY">Doodling in Math Class: Stars</a>, Video (2010).
%H A055684 Hugo Pfoertner, <a href="/A055684/a055684.pdf">Star-shaped regular polygons up to n=25.</a>
%H A055684 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StarPolygon.html">Star Polygon</a>
%F A055684 a(n) = A023022(n) - 1.
%F A055684 a(n) + A082023(n) = A140106(n). - _Gus Wiseman_, Oct 28 2020
%e A055684 The first star has five points and is unique. The next is the seven pointed star and it comes in two varieties.
%e A055684 From _Gus Wiseman_, Oct 28 2020: (Start)
%e A055684 The a(5) = 1 through a(17) = 7 irreducible pairs > 1 (shown as fractions, empty column indicated by dot):
%e A055684   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 A055684           3/4       4/5       3/8       3/10  5/9   4/11  5/11  3/14
%e A055684                               4/7       4/9         7/8   7/9   4/13
%e A055684                               5/6       5/8                     5/12
%e A055684                                         6/7                     6/11
%e A055684                                                                 7/10
%e A055684                                                                 8/9
%e A055684 (End)
%p A055684 with(numtheory): A055684 := n->(phi(n)-2)/2; seq(A055684(n), n=3..100);
%t A055684 Table[(EulerPhi[n]-2)/2, {n, 3, 50}]
%t A055684 Table[Length[Select[IntegerPartitions[n,{2}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}] (* _Gus Wiseman_, Oct 28 2020 *)
%Y A055684 Cf. A023022.
%Y A055684 Cf. A053669 smallest skip increment, A102302 skip increment of densest star polygon.
%Y A055684 A055684*2 is the ordered version.
%Y A055684 A082023 counts the complement (reducible pairs > 1).
%Y A055684 A220377, A337563, and A338332 count triples instead of pairs.
%Y A055684 A000837 counts relatively prime partitions, with strict case A078374.
%Y A055684 A002865 counts partitions with no 1's, with strict case A025147.
%Y A055684 A007359 and A337485 count pairwise coprime partitions with no 1's.
%Y A055684 A302698 counts relatively prime partitions with no 1's, with strict case A337452.
%Y A055684 A327516 counts pairwise coprime partitions, with strict case A305713.
%Y A055684 A337450 counts relatively prime compositions with no 1's, with strict case A337451.
%Y A055684 Cf. A001399, A101268, A140106, A337461, A337462, A338333.
%K A055684 nonn,easy
%O A055684 3,5
%A A055684 _Robert G. Wilson v_, Jun 09 2000