A046524 - OEIS

%I #31 Mar 13 2018 04:31:10

%S 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,

%T 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,

%U 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

%N Number of coverings of Klein bottle with n lists.

%H T. D. Noe, <a href="/A046524/b046524.txt">Table of n, a(n) for n=1..1000</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 A. D. Mednykh, <a href="https://doi.org/10.1080/00927878808823684">On the number of subgroups in the fundamental group of a closed surface</a>, Commun. in Algebra, 16, No 10 (1988), 2137-2148.

%F a(n)=d(n) (the number of divisors) for odd n.

%F 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).

%F 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

%p 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

%t 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 *)

%o (Sage)

%o def A046524(n) :

%o     f = lambda n : 1 if n % 2 == 1 else (n+7)//4

%o     return add(f(d) for d in divisors(n))

%o [A046524(n) for n in (1..87)] # _Peter Luschny_, Jul 23 2012

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

%K nonn,easy,nice

%O 1,2

%A _Valery A. Liskovets_

%E More terms from _Vladeta Jovovic_, Feb 03 2003