%I #44 Jan 19 2022 14:30:48

%S 1,3,3,6,4,9,5,11,8,12,7,19,8,15,14,20,10,24,11,26,18,21,13,37,17,24,

%T 22,33,16,42,17,37,26,30,26,53,20,33,30,52,22,54,23,47,42,39,25,71,30,

%U 51,38,54,28,66,38,67,42,48,31,94,32,51,55,70,44,78,35,68,50,78,37,108

%N Number of pairs (p,q), 0<=p<=q, such that p+q divides n.

%C Also number of orientable coverings of the Klein bottle with 2n lists (orientable m-list coverings exist only for even m).

%C Equals row sums of triangle A178650. - _Gary W. Adamson_, May 31 2010

%C Also number of inequivalent sublattices of index n of the rectangular lattice, that has the p2mm (pmm) symmetry group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A145394 (p6), A003051 (p6mm). - _Andrey Zabolotskiy_, Mar 12 2018

%H Andrey Zabolotskiy, <a href="/A069734/b069734.txt">Table of n, a(n) for n = 1..10000</a>

%H V. A. Liskovets and A. Mednykh, <a href="https://www.researchgate.net/publication/251203042">Number of non-orientable coverings of the Klein bottle</a>

%H John S. Rutherford, <a href="http://dx.doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163. [See Table 2]. [From _N. J. A. Sloane_, Feb 23 2009]

%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>

%F a(n) = A046524(2n) - A069733(2n).

%F Inverse Moebius transform of: 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... G.f.: Sum_{n>0} x^n*(1+x^n-x^(2*n))/(1-x^(2*n))/(1-x^n). - _Vladeta Jovovic_, Feb 03 2003

%F a(n) = (A000203(n) + A069735(n))/2. [Rutherford] - _N. J. A. Sloane_, Mar 13 2009

%F a(n) = Sum_{ m: m^2|n } A304182(n/m^2) + A304183(n/m^2) = A069735(n) + Sum_{ m: m^2|n } A304183(n/m^2). - _Andrey Zabolotskiy_, May 07 2018

%F a(n) = Sum_{ d|n } A008619(d) = Sum_{ d|n } (1 + floor(d/2)). - _Andrey Zabolotskiy_, Jul 20 2019

%F a(n) = (A007503(n) + A183063(n))/2. - _Peter Luschny_, Jul 20 2019

%e There are 9 pairs (p,q), 0<=p<=q, such that p+q divides 6: (0,1), (0,2), (0,3), (0,6), (1,1), (1, 2), (1, 5), (2, 4), (3, 3); thus a(6) = 9.

%e x + 3*x^2 + 3*x^3 + 6*x^4 + 4*x^5 + 9*x^6 + 5*x^7 + 11*x^8 + 8*x^9 + ...

%p with(numtheory): a := n -> (sigma(n) + tau(n) + `if`(irem(n,2) = 1, 0, tau(n/2)))/2: seq(a(n), n=1..72); # _Peter Luschny_, Jul 20 2019

%t a[n_] := (DivisorSigma[1, n] + DivisorSigma[0, n] + If[OddQ[n], 0, DivisorSigma[0, n/2]])/2;

%t Array[a, 72] (* _Jean-François Alcover_, Aug 27 2019, from Maple *)

%o (PARI) {a(n) = if( n<1, 0, sum( k=1, n, sum( j=0, k, n%(j+k) == 0)))} /* _Michael Somos_, Mar 24 2012 */

%Y Cf. A069733, A069735, A046524, A178650.

%Y Cf. A000203, A145391-A145394, A003051.

%Y Cf. A304182, A304183, A008619, A007503, A183063.

%K easy,nonn

%O 1,2

%A _Valery A. Liskovets_, Apr 07 2002

%E New description from _Vladeta Jovovic_, Feb 03 2003