%I A001001
%S A001001 1,7,13,35,31,91,57,155,130,217,133,455,183,399,403,651,307,
%T A001001 910,381,1085,741,931,553,2015,806,1281,1210,1995,871,2821,
%U A001001 993,2667,1729,2149,1767,4550,1407,2667,2379,4805,1723,5187
%N A001001 Number of sublattices of index n in generic 3-dimensional lattice.
%C A001001 These sublattices are in 1-1 correspondence with matrices
%C A001001 [a b d]
%C A001001 [0 c e]
%C A001001 [0 0 f]
%C A001001 with acf = n, b = 0..c-1, d = 0..f-1, e = 0..f-1. The sublattice is primitive 
               if gcd(a,b,c,d,e,f) = 1.
%C A001001 Equals row sums of triangle A127108. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jul 27 2008
%D A001001 M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., 
               Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
%D A001001 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.
%D A001001 J. S. Rutherford, The enumeration and symmetry-significant properties 
               of derivative lattices, Acta Cryst. A48 (1992), 500-508. [From N. 
               J. A. Sloane, Mar 14 2009]
%D A001001 J. S. Rutherford, The enumeration and symmetry-significant properties 
               of derivative lattices II, Acta Cryst. A49 (1993), 293-300. [From 
               N. J. A. Sloane, Mar 14 2009]
%D A001001 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 5.13(d), pp. 76 and 113.
%H A001001 T. D. Noe, <a href="b001001.txt">Table of n, a(n) for n=1..1000</a>
%H A001001 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A001001 <a href="Sindx_Su.html#sublatts">Index entries for sequences related 
               to sublattices</a>
%F A001001 If n = Product p^m, a(n) = Product (p^(m + 1) - 1)(p^(m + 2) - 1)/(p 
               - 1)(p^2 - 1). Or, a(n) = Sum_{d}n} sigma(n/d)*d^2.
%F A001001 a(n) = Sum_{d|n} d*sigma(d). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 06 2001
%F A001001 Multiplicative with a(p^e) = ((p^(e+1)-1)(p^(e+2)-1))/((p-1)(p^2-1)). 
               - David W. Wilson, Sep 01, 2001
%F A001001 Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)
%o A001001 (PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
%o A001001 N=17; default(seriesprecision,N); x=z+O(z^(N+1))
%o A001001 c=sum(j=1,N,j*x^j);
%o A001001 t=1/prod(j=1,N, eta(x^(j))^j)
%o A001001 t=log(t)
%o A001001 t=serconvol(t,c)
%o A001001 Vec(t)
%Y A001001 Cf. A060983.
%Y A001001 Cf. A127108.
%Y A001001 Primes in this sequence are in A053183.
%Y A001001 Sequence in context: A026318 A061204 A060983 this_sequence A067692 A117706 
               A066673
%Y A001001 Adjacent sequences: A000998 A000999 A001000 this_sequence A001002 A001003 
               A001004
%K A001001 nonn,easy,nice,mult
%O A001001 1,2
%A A001001 N. J. A. Sloane (njas(AT)research.att.com).