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A023645 a(n) = tau(n)-1 if n is odd or tau(n)-2 if n is even. 24
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
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
REFERENCES
CRC Handbook of Combinatorial Designs, 1996, p. 649.
LINKS
Felix Fröhlich et al., Rings of regular polygons, SeqFan thread, March 26 2017.
Gordon Royle, Transitive Graphs
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
EXAMPLE
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 + ...
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 *)
PROG
(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
CROSSREFS
Second column of A072528.
Sequence in context: A158946 A303428 A223853 * A167865 A218654 A054571
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Dec 03 2001
STATUS
approved

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Last modified May 30 13:54 EDT 2023. Contains 363050 sequences. (Running on oeis4.)