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A001001 Number of sublattices of index n in generic 3-dimensional lattice. 38
1, 7, 13, 35, 31, 91, 57, 155, 130, 217, 133, 455, 183, 399, 403, 651, 307, 910, 381, 1085, 741, 931, 553, 2015, 806, 1281, 1210, 1995, 871, 2821, 993, 2667, 1729, 2149, 1767, 4550, 1407, 2667, 2379, 4805, 1723, 5187, 1893, 4655, 4030, 3871, 2257, 8463, 2850, 5642, 3991, 6405, 2863 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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, Jul 27 2008

REFERENCES

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 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.

LINKS

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

M. Baake, N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.

J. Liouville, Théorème concernant les sommes de diviseurs des nombres, Journal de mathématiques pures et appliquées 2e série, tome 2 (1857), p. 56-.

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

J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Acryst. (1992) A48, 500-508

Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013.

Index entries for sequences related to sublattices

FORMULA

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, Dirichlet convolution of A000290 and A000203.

a(n) = Sum_{d|n} d*sigma(d). - Vladeta Jovovic, 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).

L.g.f.: -log(Product_{k>=1} (1 - x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018

MAPLE

nmax := 100:

L12 := [seq(1, i=1..nmax) ];

L27 := [seq(i, i=1..nmax) ];

L290 := [seq(i^2, i=1..nmax) ];

DIRICHLET(L12, L27) ;

DIRICHLET(%, L290) ; # R. J. Mathar, Sep 25 2017

MATHEMATICA

a[n_] := Sum[ d*DivisorSigma[1, d], {d, Divisors[n]}]; Table[ a[n], {n, 1, 42}] (* Jean-François Alcover, Jan 20 2012, after Vladeta Jovovic *)

PROG

(PARI)

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)

/* Joerg Arndt, May 03 2008 */

(PARI) a(n)=sumdiv(n, d, d * sumdiv(d, t, t ) );  /* Joerg Arndt, Oct 07 2012 */

(PARI) a(n)=sumdivmult(n, d, sigma(d)*d) \\ Charles R Greathouse IV, Sep 09 2014

CROSSREFS

Cf. A060983, A064987 (Mobius transform).

Cf. A061256, A127108, A226313, A301777.

Primes in this sequence are in A053183.

Sequence in context: A026318 A061204 A060983 * A067692 A117706 A066673

Adjacent sequences:  A000998 A000999 A001000 * A001002 A001003 A001004

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 20 20:29 EDT 2018. Contains 315241 sequences. (Running on oeis4.)