cp's OEIS Frontend

A027844 Number of subgroups of index n of fundamental group of the non-orientable cycle bundle over the Klein bottle.

%I A027844 #28 Dec 16 2021 22:47:49
%S A027844 1,7,7,27,11,55,15,91,34,97,23,231,27,147,77,299,35,334,39,437,105,
%T A027844 271,47,847,86,345,142,699,59,865,63,1003,161,517,165,1590,75,615,189,
%U A027844 1701,83,1371,87,1391,374,835,95,3023,162,1322,245,1821,107,2062,253,2835
%N A027844 Number of subgroups of index n of fundamental group of the non-orientable cycle bundle over the Klein bottle.
%C A027844 From a recent general formula of Stanley's for the number of subgroups in G\times Z.
%D A027844 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64.
%H A027844 Gheorghe Coserea, <a href="/A027844/b027844.txt">Table of n, a(n) for n = 1..20000</a>
%H A027844 G. Chelnokov, M. Deryagina, A. Mednykh, <a href="http://arxiv.org/abs/1502.01528">On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2</a>, arXiv preprint arXiv:1502.01528 [math.AT], 2015.
%H A027844 V. A. Liskovets and A. Mednykh, <a href="http://dx.doi.org/10.1080/00927870008826924">Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces</a>, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
%F A027844 Sum k*b(k), k|n, where b(k) is the number of n-list coverings of the Klein bottle (A046524).
%t A027844 b[k_] := If[OddQ[k], DivisorSigma[0, k], (3 DivisorSigma[0, k] + DivisorSigma[1, k/2] - DivisorSigma[0, k/2])/2]; a[n_] := Sum[k*b[k], {k, Divisors[n]}]; Table[a[n], {n, 1, 56}] (* _Jean-François Alcover_, Jul 19 2012 *)
%o A027844 (PARI)
%o A027844 A001001(n) = sumdiv(n, d, sigma(d) * d);
%o A027844 A060640(n) = sumdiv(n, d, sigma(n\d) * d);
%o A027844 S1(n)      = if (n%2, 0, A001001(n\2));
%o A027844 S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
%o A027844 S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
%o A027844 a(n) = S1(n) + S11(n) + S21(n);
%o A027844 vector(56, n, a(n))  \\ _Gheorghe Coserea_, May 05 2016
%Y A027844 Cf. A001001, A027845, A046524.
%K A027844 nonn,easy,nice
%O A027844 1,2
%A A027844 _N. J. A. Sloane_
%E A027844 More terms from _Valery A. Liskovets_
%E A027844 Corrected and extended by _Vladeta Jovovic_, Feb 03 2003