A069734 Number of pairs (p,q), 0<=p<=q, such that p+q divides n.

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, 22, 33, 16, 42, 17, 37, 26, 30, 26, 53, 20, 33, 30, 52, 22, 54, 23, 47, 42, 39, 25, 71, 30, 51, 38, 54, 28, 66, 38, 67, 42, 48, 31, 94, 32, 51, 55, 70, 44, 78, 35, 68, 50, 78, 37, 108

Offset: 1

Author

Valery A. Liskovets, Apr 07 2002

Keywords

• easy
• nonn

Comments

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

Equals row sums of triangle A178650. - Gary W. Adamson, May 31 2010

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

Examples

			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.
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 + ...
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
• V. A. Liskovets and A. Mednykh, Number of non-orientable coverings of the Klein bottle
• John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2]. [From _N. J. A. Sloane_, Feb 23 2009]
• Index entries for sequences related to sublattices

Crossrefs

Cf. A069733, A069735, A046524, A178650.

Cf. A000203, A145391-A145394, A003051.

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

Programs

Maple

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

Mathematica

a[n_] := (DivisorSigma[1, n] + DivisorSigma[0, n] + If[OddQ[n], 0, DivisorSigma[0, n/2]])/2;
Array[a, 72] (* Jean-François Alcover, Aug 27 2019, from Maple *)

PARI

{a(n) = if( n<1, 0, sum( k=1, n, sum( j=0, k, n%(j+k) == 0)))} /* Michael Somos, Mar 24 2012 */

Formula

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

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

a(n) = (A000203(n) + A069735(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009

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

a(n) = Sum_{ d|n } A008619(d) = Sum_{ d|n } (1 + floor(d/2)). - Andrey Zabolotskiy, Jul 20 2019

a(n) = (A007503(n) + A183063(n))/2. - Peter Luschny, Jul 20 2019

Extensions

New description from Vladeta Jovovic, Feb 03 2003