

A343005


a(n) is the number of dihedral symmetries D_{2m} (m >= 3) that configurations of n nonoverlapping equal circles can possess.


2



0, 1, 2, 2, 3, 3, 3, 4, 4, 3, 5, 5, 3, 5, 6, 4, 5, 5, 5, 7, 5, 3, 7, 8, 4, 5, 7, 5, 7, 7, 5, 7, 5, 5, 10, 8, 3, 5, 9, 7, 7, 7, 5, 9, 7, 3, 9, 10, 6, 7, 7, 5, 7, 9, 9, 9, 5, 3, 11, 11, 3, 7, 10, 8, 9, 7, 5, 7, 9, 7, 11, 11, 3, 7, 9, 7, 9, 7, 9, 12, 6, 3, 11, 13
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

2,3


LINKS

Table of n, a(n) for n=2..85.


FORMULA

For n >= 2, a(n) = A274010(n)  1 = A023645(n) + A023645(n1) = tau(n) + tau(n1)  3, where tau(n) = A000005(n), the number of divisors of n.


EXAMPLE

a(2) = 0 because the configuration of 2 circles only possesses D_{4} symmetry.
a(6) = 3 because configurations of 6 circles can have three dihedral symmetries: D_{12} (6 circles arranged in regular hexagon shape), D_{10} (5 circles arranged in regular pentagon shape and the other circle in the center of the pentagon), and D_{6} (6 circles arranged in equilateral triangle shape).


MATHEMATICA

Table[DivisorSigma[0, n]+DivisorSigma[0, n1]3, {n, 2, 85}] (* Stefano Spezia, Apr 06 2021 *)


PROG

(Python)
from sympy import divisor_count
for n in range(2, 101): print(divisor_count(n) + divisor_count(n  1))  3)


CROSSREFS

Cf. A000005, A023645, A274010.
Sequence in context: A179864 A070082 A085727 * A143442 A137300 A201052
Adjacent sequences: A343002 A343003 A343004 * A343006 A343007 A343008


KEYWORD

nonn


AUTHOR

YaPing Lu, Apr 02 2021


STATUS

approved



