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

Author

Valery A. Liskovets

Keywords

• nonn
• easy
• nice

Links

• 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.

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Sage

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

Formula

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

Extensions

More terms from Vladeta Jovovic, Feb 03 2003