A023645 a(n) = tau(n)-1 if n is odd or tau(n)-2 if n is even.

0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 3, 1, 4, 1, 4, 3, 2, 1, 6, 2, 2, 3, 4, 1, 6, 1, 4, 3, 2, 3, 7, 1, 2, 3, 6, 1, 6, 1, 4, 5, 2, 1, 8, 2, 4, 3, 4, 1, 6, 3, 6, 3, 2, 1, 10, 1, 2, 5, 5, 3, 6, 1, 4, 3, 6, 1, 10, 1, 2, 5, 4, 3, 6, 1, 8, 4, 2, 1, 10, 3, 2, 3, 6, 1, 10, 3, 4, 3, 2, 3, 10, 1, 4, 5, 7, 1, 6, 1, 6

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Vertex-transitive graphs of valency 2 with n nodes.

Number of values of k such that n+2 divided by k leaves a remainder 2. - Amarnath Murthy, Aug 01 2002

Number of divisors of n that are less than n/2. - Peter Munn, Mar 31 2017, or equivalently, number of divisors of n that are greater than 2. - Antti Karttunen, Feb 20 2023

For n > 2, a(n) is the number of planar arrangements of equal-sized regular n-gons such that their centers lie on a circle and neighboring n-gons have an edge in common. - Peter Munn, Apr 23 2017

Number of partitions of n into two distinct parts such that the smaller divides the larger. - Wesley Ivan Hurt, Dec 21 2017

Examples

			x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + x^11 + 4*x^12 + ...
		

References

• CRC Handbook of Combinatorial Designs, 1996, p. 649.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Felix Fröhlich et al., Rings of regular polygons, SeqFan thread, March 26 2017.
• Gordon Royle, Transitive Graphs

Crossrefs

Cf. A000005, A001620, A023637-A023647.

Cf. A014405, A049992, A069930, A095374, A296955, A321014.

Second column of A072528.

Programs

Maple

with(numtheory); f := n->if n mod 2 = 1 then tau(n)-1 else tau(n)-2; fi;

Mathematica

Table[s = DivisorSigma[0, n]; If[OddQ[n], s - 1, s - 2], {n, 100}] (* T. D. Noe, Nov 18 2013 *)
Array[DivisorSigma[0, #] - 1 - Boole@ EvenQ@ # &, 104] (* Michael De Vlieger, Apr 25 2017 *)

PARI

{a(n) = if( n<1, 0, numdiv(n) - 2 + n%2)} /* Michael Somos, Apr 29 2003 */

PARI

a(n) = sumdiv(n, d, d < n/2); \\ Michel Marcus, Apr 01 2017

Formula

G.f.: Sum_{k>0} x^(3*k) / (1 - x^k). - Michael Somos, Apr 29 2003.

a(2*n) = A069930(n). a(2*n + 1) = A095374(n). - Michael Somos, Aug 30 2012

a(n) = A072528(n+2,2) for n > 2. - Peter Munn, May 14 2017

From Peter Bala, Jan 13 2021: (Start)

a(n) = Sum_{ d|n, d < n/2 } 1. Cf. A296955.

G.f.: Sum_{k >= 3} x^k/(1 - x^k). (End)

a(n) = A049992(n) - A014405(n). - Antti Karttunen, Feb 20 2023

Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 5/2), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

Extensions

More terms from Vladeta Jovovic, Dec 03 2001