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

1, 7, 7, 27, 11, 55, 15, 91, 34, 97, 23, 231, 27, 147, 77, 299, 35, 334, 39, 437, 105, 271, 47, 847, 86, 345, 142, 699, 59, 865, 63, 1003, 161, 517, 165, 1590, 75, 615, 189, 1701, 83, 1371, 87, 1391, 374, 835, 95, 3023, 162, 1322, 245, 1821, 107, 2062, 253, 2835

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

From a recent general formula of Stanley's for the number of subgroups in G\times Z.

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64.

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..20000
• G. Chelnokov, M. Deryagina, A. Mednykh, On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2, arXiv preprint arXiv:1502.01528 [math.AT], 2015.
• V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.

Crossrefs

Cf. A001001, A027845, A046524.

Programs

Mathematica

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 *)

PARI

A001001(n) = sumdiv(n, d, sigma(d) * d);
A060640(n) = sumdiv(n, d, sigma(n\d) * d);
S1(n)      = if (n%2, 0, A001001(n\2));
S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
a(n) = S1(n) + S11(n) + S21(n);
vector(56, n, a(n))  \\ Gheorghe Coserea, May 05 2016

Formula

Sum k*b(k), k|n, where b(k) is the number of n-list coverings of the Klein bottle (A046524).

Extensions

More terms from Valery A. Liskovets

Corrected and extended by Vladeta Jovovic, Feb 03 2003