|
|
A001001
|
|
Number of sublattices of index n in generic 3-dimensional lattice.
|
|
50
|
|
|
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.
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
|
|
|
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.
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
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2*zeta(3)/18 = 0.659101... . - Amiram Eldar, Oct 19 2022
|
|
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) ;
|
|
MATHEMATICA
|
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 *)
|
|
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)
(PARI) a(n)=sumdiv(n, d, d * sumdiv(d, t, t ) ); /* Joerg Arndt, Oct 07 2012 */
|
|
CROSSREFS
|
Primes in this sequence are in A053183.
|
|
KEYWORD
|
nonn,easy,nice,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|