cp's OEIS Frontend

A069735 Number of regular orientable coverings of the Klein bottle with 2n lists.

%I A069735 #64 Dec 22 2023 05:56:47
%S A069735 1,3,2,5,2,6,2,7,3,6,2,10,2,6,4,9,2,9,2,10,4,6,2,14,3,6,4,10,2,12,2,
%T A069735 11,4,6,4,15,2,6,4,14,2,12,2,10,6,6,2,18,3,9,4,10,2,12,4,14,4,6,2,20,
%U A069735 2,6,6,13,4,12,2,10,4,12,2,21,2,6,6,10,4,12,2,18,5,6,2,20,4,6,4,14,2,18
%N A069735 Number of regular orientable coverings of the Klein bottle with 2n lists.
%C A069735 Dirichlet convolution of A000012 by A040001. - _R. J. Mathar_, Mar 30 2011
%C A069735 a(n) is the number of full-dimensional lattices with volume n in Z^2 which are symmetric about a coordinate axis (equivalently, about both). - _Álvar Ibeas_, Mar 19 2021
%H A069735 Andrey Zabolotskiy, <a href="/A069735/b069735.txt">Table of n, a(n) for n = 1..10000</a>
%H A069735 Valery A. Liskovets and Alexander Mednykh, <a href="https://www.researchgate.net/publication/251203042">Number of non-orientable coverings of the Klein bottle</a>, 2002.
%H A069735 John S. Rutherford, <a href="http://dx.doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009
%F A069735 Multiplicative with a(2^e)=2e+1 and a(p^e)=e+1 for e>0 and an odd prime p.
%F A069735 a(n) = d(n)+d(n/2) for even n and a(n) = d(n) otherwise where d(n) is the number of divisors of n (A000005).
%F A069735 G.f.: Sum_{k>0} x^k*(1+2*x^k)/(1-x^(2*k)). - _Vladeta Jovovic_, Dec 16 2002
%F A069735 Dirichlet g.f.: (1+2^(-s))*zeta^2(s) [ Rutherford]. - _N. J. A. Sloane_, Feb 23 2009
%F A069735 Moebius transform is period 2 sequence [ 1, 2, ...]. - _Michael Somos_, Mar 24 2012
%F A069735 a(2*n - 1) = A099774(n).
%F A069735 a(n) = Sum_{ m: m^2|n } A304182(n/m^2). - _Andrey Zabolotskiy_, May 07 2018
%F A069735 Sum_{k=1..n} a(k) ~ 3*n*log(n)/2 + (3*gamma - 3/2 - log(2)/2)*n, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 04 2019
%F A069735 a(n) = 3*tau(n) - tau(2*n). - _Ridouane Oudra_, Mar 15 2021
%F A069735 a(n) = A320111(n) + (A059841(n)*A000005(n)), i.e. a(n) = A320111(n) if n is odd, and a(n) = A320111(n) + A000005(n) if n is even. - _Antti Karttunen_, Mar 17 2021
%F A069735 a(n) = A000005(n) + A183063(n) = 2*A000005(n) - A001227(n). - _Amiram Eldar_, Dec 22 2023
%e A069735 x + 3*x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 6*x^6 + 2*x^7 + 7*x^8 + 3*x^9 + 6*x^10 + ...
%p A069735 read("transforms") : nmax := 100 :
%p A069735 L := [1,1,seq(0,i=1..nmax)] :
%p A069735 MOBIUSi(%) :
%p A069735 MOBIUSi(%) ; # _R. J. Mathar_, Sep 25 2017
%p A069735 with(NumberTheory): seq(tau(n) + `if`(n::odd, 0, tau(n/2)), n=1..100); # _Peter Luschny_, Mar 19 2021
%t A069735 d[n_] := DivisorSigma[0, n];
%t A069735 a[n_] := If[EvenQ[n], d[n] + d[n/2], d[n]];
%t A069735 Array[a, 100] (* _Jean-François Alcover_, Aug 27 2019 *)
%o A069735 (PARI) {a(n) = if( n<1, 0, numdiv(n) + if( n%2, 0, numdiv( n / 2)))} /* _Michael Somos_, Mar 24 2012 */
%Y A069735 Equals row sums of triangle A143110. - _Gary W. Adamson_, Jul 25 2008
%Y A069735 Cf. A000005, A000012, A001227, A001620, A040001, A059841, A069734, A099774, A183063, A304182, A320111.
%K A069735 mult,easy,nonn
%O A069735 1,2
%A A069735 _Valery A. Liskovets_, Apr 07 2002
%E A069735 Corrected by _T. D. Noe_, Nov 13 2006