A069734 -id:A069734 - cp's OEIS frontend

A046524 Number of coverings of Klein bottle with n lists.

1, 3, 2, 5, 2, 7, 2, 8, 3, 8, 2, 13, 2, 9, 4, 13, 2, 14, 2, 16, 4, 11, 2, 23, 3, 12, 4, 19, 2, 22, 2, 22, 4, 14, 4, 30, 2, 15, 4, 30, 2, 26, 2, 25, 6, 17, 2, 41, 3, 23, 4, 28, 2, 30, 4, 37, 4, 20, 2, 50, 2, 21, 6, 39, 4, 34, 2, 34, 4, 34, 2, 59, 2, 24, 6, 37, 4, 38, 2, 56, 5, 26, 2, 62, 4, 27, 4

Offset: 1

Valery A. Liskovets

T. D. Noe, Table of n, a(n) for n=1..1000
V. A. Liskovets and A. Mednykh, Number of non-orientable coverings of the Klein bottle
A. D. Mednykh, On the number of subgroups in the fundamental group of a closed surface, Commun. in Algebra, 16, No 10 (1988), 2137-2148.

Cf. A027842, A027844, A000005, A000203, A069733, A069734, A069735.

with(numtheory); A046524:=n->`if`(type(n/2, integer), (3*tau(n) + sigma(n/2) - tau(n/2))/2, tau(n)); seq(A046524(n), n=1..100); # Wesley Ivan Hurt, Feb 14 2014

kb[n_]:=If[OddQ[n],DivisorSigma[0,n],(3DivisorSigma[0,n]+ DivisorSigma[ 1,n/2]- DivisorSigma[0,n/2])/2]; Array[kb,90] (* Harvey P. Dale, Oct 08 2011 *)

def A046524(n) :
    f = lambda n : 1 if n % 2 == 1 else (n+7)//4
    return add(f(d) for d in divisors(n))
[A046524(n) for n in (1..87)] # Peter Luschny, Jul 23 2012

a(n)=d(n) (the number of divisors) for odd n.
a(n)=[3d(n)+sigma(n/2)-d(n/2)]/2 for even n where d(n) is the number and sigma(n) the sum of divisors of n (A000005 and A000203).
Inverse Moebius transform of 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 4, 1, 5, 1, 5, 1, 6, 1, 6, 1, 7, 1, 7, ... . G.f.: Sum_{n>1} x^n*(1+2*x^n-x^(4*n)-x^(5*n))/(1+x^(2*n))/(1-x^(2*n))^2. - Vladeta Jovovic, Feb 03 2003

More terms from Vladeta Jovovic, Feb 03 2003

A304183 Number of primitive inequivalent oblique sublattices of rectangular lattice of index n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 5, 8, 6, 9, 10, 10, 8, 15, 9, 14, 14, 15, 11, 20, 14, 18, 17, 20, 14, 30, 15, 22, 22, 24, 22, 32, 18, 27, 26, 32, 20, 42, 21, 32, 34, 33, 23, 44, 27, 42, 34, 38, 26, 51, 34, 44, 38, 42, 29, 64, 30, 45, 46, 46, 40, 66, 33, 50, 46

Offset: 1

Andrey Zabolotskiy, May 07 2018

nonn

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 4. Contains errors for n = 24 and 28.]

Cf. A304182 (primitive mirror-symmetric sublattices), A069734 (all sublattices), A001615 (primitive sublattices of oblique lattice).

a(n) = (A001615(n) - A304182(n))/2.

A178650 Triangle read by rows A051731 * A070909.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 3, 1, 1, 1, 2, 0, 1, 0, 1, 4, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 4, 1, 2, 1, 1, 0, 1, 1, 3, 0, 2, 0, 1, 0, 1, 0, 1, 4, 1, 2, 0, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 6, 1, 4, 1, 2, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Gary W. Adamson, May 31 2010

Row sums = A069734: (1, 3, 3, 6, 4, 9, 5, 11,...)

Left border = d(n), A000005

			First few rows of the triangle =
1;
2, 1;
2, 0, 1;
3, 1, 1, 1;
2, 0, 1, 0, 1;
4, 1, 2, 0, 1, 1;
2, 0, 1, 0, 1, 0, 1;
4, 1, 2, 1, 1, 0, 1, 1;
3, 0, 2, 0, 1, 0, 1, 0, 1;
4, 1, 2, 0, 2, 0, 1, 0, 1, 1;
2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
6, 1, 4, 1, 2, 1, 1, 0, 1, 0, 1, 1;
2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
4, 1, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 1;
4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1
5, 1, 3, 1, 2, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1;
2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
...

Cf. A051731, A070909, A069734.

As infinite lower triangular matrices, A051731 * A070909; where A051731 = the inverse Mobius transform.

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A046524 Number of coverings of Klein bottle with n lists.

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A304183 Number of primitive inequivalent oblique sublattices of rectangular lattice of index n.

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A178650 Triangle read by rows A051731 * A070909.

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