A343005 a(n) is the number of dihedral symmetries D_{2m} (m >= 3) that configurations of n non-overlapping equal circles can possess.

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

Offset: 2

Author

Ya-Ping Lu, Apr 02 2021

Keywords

• nonn

Examples

			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).
		

Crossrefs

Cf. A000005, A023645, A274010.

Programs

Mathematica

Table[DivisorSigma[0,n]+DivisorSigma[0,n-1]-3,{n,2,85}] (* Stefano Spezia, Apr 06 2021 *)

Python

from sympy import divisor_count
for n in range(2, 101):
    print(divisor_count(n) + divisor_count(n - 1) - 3, end=", ")

Formula

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