1,2

These sublattices are in 1-1 correspondence with matrices

[a b d]

[0 c e]

[0 0 f]

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.

Equals row sums of triangle A127108. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2008

M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

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.

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]

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]

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(d), pp. 76 and 113.

T. D. Noe, Table of n, a(n) for n=1..1000

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Index entries for sequences related to sublattices

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.

a(n) = Sum_{d|n} d*sigma(d). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 06 2001

Multiplicative with a(p^e) = ((p^(e+1)-1)(p^(e+2)-1))/((p-1)(p^2-1)). - David W. Wilson, Sep 01, 2001

Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)

(PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)

N=17; default(seriesprecision, N); x=z+O(z^(N+1))

c=sum(j=1, N, j*x^j);

t=1/prod(j=1, N, eta(x^(j))^j)

t=log(t)

t=serconvol(t, c)

Vec(t)

Cf. A060983.

Cf. A127108.

Primes in this sequence are in A053183.

Sequence in context: A026318 A061204 A060983 this_sequence A067692 A117706 A066673

Adjacent sequences: A000998 A000999 A001000 this_sequence A001002 A001003 A001004

nonn,easy,nice,mult

N. J. A. Sloane (njas(AT)research.att.com).