A389108 Number of 4 X 4 matrices in Hermite normal form with determinant n.

1, 8, 27, 96, 125, 324, 343, 832, 972, 1500, 1331, 3384, 2197, 4116, 4500, 7744, 4913, 11178, 6859, 15000, 12348, 15972, 12167, 31896, 18750, 26364, 26973, 41160, 24389, 57150, 29791, 64000, 47916, 58956, 51450, 116910, 50653, 82308, 79092, 137800, 68921, 154350

Offset

1,2

Comments

a(n) is also the number of lattices in R^4 with determinant n.

Links

Table of n, a(n) for n=1..42.

Wikipedia, Hermite normal form

Formula

a(n) = Sum_{ d_1 * d_2 * d_3 * d_4 = n ; d_1 <= d_2 <= d_3 <= d_4 } d_2 * d_3^2 * d_4^3.

For p prime: a(p) = p^3.

Example

For n=2, there are 8 4 X 4 matrices M in Hermite normal form (rows convention) with det(M) = 2:
  {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 2}},
  {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 1}, {0, 0, 0, 2}},
  {{1, 0, 0, 0}, {0, 1, 0, 1}, {0, 0, 1, 0}, {0, 0, 0, 2}},
  {{1, 0, 0, 0}, {0, 1, 0, 1}, {0, 0, 1, 1}, {0, 0, 0, 2}},
  {{1, 0, 0, 1}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 2}},
  {{1, 0, 0, 1}, {0, 1, 0, 0}, {0, 0, 1, 1}, {0, 0, 0, 2}},
  {{1, 0, 0, 1}, {0, 1, 0, 1}, {0, 0, 1, 0}, {0, 0, 0, 2}},
  {{1, 0, 0, 1}, {0, 1, 0, 1}, {0, 0, 1, 1}, {0, 0, 0, 2}}

Mathematica

a[n_] :=Module[{factors = Union[Sort/@Select[Tuples[Divisors[n], 4], Times @@ # == n &]]}, Sum[Product[ factors[[i, j]]^(j - 1) , {j, 4 }] , {i, Length[factors]}]]

Crossrefs

Sequence in context: A383747 A343323 A079712 * A056598 A343729 A343283

Adjacent sequences: A389105 A389106 A389107 * A389109 A389110 A389111

Keyword

nonn

Author

Andrew Ortegaray, Sep 23 2025

Status

approved