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

2,3

Links

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

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.

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, n-1]-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, end=", ")

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

Ya-Ping Lu, Apr 02 2021

Status

approved