A001001 Number of sublattices of index n in generic 3-dimensional lattice.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• mult

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.

Total area of all distinct rectangles whose side lengths are divisors of n, and whose length is an integer multiple of the width. - Wesley Ivan Hurt, Aug 23 2020

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Michael Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.; arXiv:math/0605222 [math.MG], 2006.
• M. Baake and 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.
• 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.
• 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. Cryst. (1992) A48, 500-508
• J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. [_N. J. A. Sloane_, Mar 14 2009]
• Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013.
• Index entries for sequences related to sublattices.

Crossrefs

Column 3 of A160870.

Cf. A060983, A064987 (Mobius transform).

Cf. A061256, A226313, A301777.

Primes in this sequence are in A053183.

Cf. A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

Programs

Haskell

a001001 n = sum [sum [k * (if k `mod` l == 0 then l else 0) | k <- [1..n], n `mod` k == 0 ] | l <- [1..n]]
a = [ a001001 n | n <- [1..53]]
putStrLn $ concat $ map (++ ", ") (map show a) -- Miles Wilson, Apr 04 2025

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 *)
f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 2}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

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

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

a(n) = Sum_{d1|n, d2|n, d1|d2} d1*d2. - Wesley Ivan Hurt, Aug 23 2020

Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2*zeta(3)/18 = 0.659101... . - Amiram Eldar, Oct 19 2022

G.f.: Sum_{k>=1} Sum {l>=1} k*l^2*x^(k*l - 1)/(1 - x^(k*l)). - Miles Wilson, Apr 04 2025