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%I #73 Jan 08 2024 01:35:51
%S 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,
%T 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,
%U 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
%N a(n) = tau(n)-1 if n is odd or tau(n)-2 if n is even.
%C Vertex-transitive graphs of valency 2 with n nodes.
%C Number of values of k such that n+2 divided by k leaves a remainder 2. - _Amarnath Murthy_, Aug 01 2002
%C 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
%C 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
%C Number of partitions of n into two distinct parts such that the smaller divides the larger. - _Wesley Ivan Hurt_, Dec 21 2017
%D CRC Handbook of Combinatorial Designs, 1996, p. 649.
%H T. D. Noe, <a href="/A023645/b023645.txt">Table of n, a(n) for n = 1..10000</a>
%H Felix Fröhlich et al., <a href="http://list.seqfan.eu/oldermail/seqfan/2017-March/017399.html">Rings of regular polygons</a>, SeqFan thread, March 26 2017.
%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/trans/">Transitive Graphs</a>
%F G.f.: Sum_{k>0} x^(3*k) / (1 - x^k). - _Michael Somos_, Apr 29 2003.
%F a(2*n) = A069930(n). a(2*n + 1) = A095374(n). - _Michael Somos_, Aug 30 2012
%F a(n) = A072528(n+2,2) for n > 2. - _Peter Munn_, May 14 2017
%F From _Peter Bala_, Jan 13 2021: (Start)
%F a(n) = Sum_{ d|n, d < n/2 } 1. Cf. A296955.
%F G.f.: Sum_{k >= 3} x^k/(1 - x^k). (End)
%F a(n) = A049992(n) - A014405(n). - _Antti Karttunen_, Feb 20 2023
%F 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
%e 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 + ...
%p with(numtheory); f := n->if n mod 2 = 1 then tau(n)-1 else tau(n)-2; fi;
%t Table[s = DivisorSigma[0, n]; If[OddQ[n], s - 1, s - 2], {n, 100}] (* _T. D. Noe_, Nov 18 2013 *)
%t Array[DivisorSigma[0, #] - 1 - Boole@ EvenQ@ # &, 104] (* _Michael De Vlieger_, Apr 25 2017 *)
%o (PARI) {a(n) = if( n<1, 0, numdiv(n) - 2 + n%2)} /* _Michael Somos_, Apr 29 2003 */
%o (PARI) a(n) = sumdiv(n, d, d < n/2); \\ _Michel Marcus_, Apr 01 2017
%Y Cf. A000005, A001620, A023637-A023647.
%Y Cf. A014405, A049992, A069930, A095374, A296955, A321014.
%Y Second column of A072528.
%K nonn,easy
%O 1,6
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Dec 03 2001
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