A054908 -id:A054908 - cp's OEIS frontend

A005134 Number of n-dimensional unimodular lattices (or quadratic forms).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 8, 9, 13, 16, 28, 40, 68, 117, 297, 665, 2566, 17059, 374062

Offset: 0

N. J. A. Sloane

King gives the lower bounds a(29) >= 37938009 and a(30) >= 20169641025. - Robin Visser, Feb 08 2025

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863. See page 854.

Cf. A054907, A054908, A054909, A054911.

If 8 divides n, then a(n) = A054911(n) + A054909(n/8), otherwise a(n) = A054911(n). - Robin Visser, Jan 24 2025

a(n) >= 2*A241121(n)/A241122(n). - Robin Visser, Feb 08 2025

a(26)-a(28) added from Bill Allombert's and Gaëtan Chenevier's computations by Robin Visser, Jan 24 2025

A054911 Number of n-dimensional odd unimodular lattices (or quadratic forms).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 9, 13, 16, 28, 40, 68, 117, 273, 665, 2566, 17059, 374062

Offset: 0

N. J. A. Sloane, May 23 2000

a(n) is also the class number of Z^n (the standard lattice with the identity as the basis), as every n-dimensional odd unimodular lattice lies in the same genus as Z^n. - Robin Visser, Jan 24 2025

King gives the lower bounds a(29) >= 37938009 and a(30) >= 20169641025. - Robin Visser, Feb 08 2025

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.

Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
Gaëtan Chenevier, Unimodular hunting, Cogent Seminar, Jul 05 2021.
Gaëtan Chenevier, Unimodular hunting, Modular Forms Workshop, Oberwolfach online, Feb 2021.
Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
Steven R. Finch, Minkowski-Siegel mass constants [Broken link]
Steven R. Finch, Minkowski-Siegel mass constants
Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863. See Table 3 page 854.

Cf. A005134, A054907, A054908, A054909, A241121, A241122.

function a(n)
    if n lt 3 then return Min(1,n); end if;
    L := NumberFieldLattice(QNF(), n);
    return #GenusRepresentatives(L);
end function;  // Robin Visser, Jan 24 2025

If 8 divides n, then a(n) = A005134(n) - A054909(n/8), otherwise a(n) = A005134(n). - Robin Visser, Jan 24 2025

a(n) >= 2*A241121(n)/A241122(n). - Robin Visser, Feb 08 2025

A054909 Number of 8n-dimensional even unimodular lattices (or quadratic forms).

Original entry on oeis.org

1, 1, 2, 24

Offset: 0

N. J. A. Sloane, May 23 2000

King shows that a(4) >= 1162109024. - Charles R Greathouse IV, Nov 05 2013

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.

Steven R. Finch, Minkowski-Siegel mass constants [Broken link]
Steven R. Finch, Minkowski-Siegel mass constants
Oliver King, A mass formula for unimodular lattices with no roots, arXiv:math/0012231 [math.NT], 2000-2001; Mathematics of Computation 72:242 (2003), pp. 839-863.

Cf. A005134, A054907, A054908, A054911.

a(n) = A005134(8*n) - A054911(8*n). - Robin Visser, Jan 24 2025

The classical mass formula shows that the next term is at least 8*10^7.

Oliver King and Richard Borcherds (reb(AT)math.berkeley.edu) have recently improved this estimate and have shown that a(4), the number in dimension 32, is at least 10^9 (Jul 22 2000)

A054907 Number of n-dimensional unimodular lattices (or quadratic forms) containing no vectors of norm 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 3, 1, 4, 3, 12, 12, 28, 49, 180, 368, 1901, 14493, 357003

Offset: 0

N. J. A. Sloane, May 23 2000

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.

Gaëtan Chenevier, Statistics for Kneser p-neighbors, arXiv:2104.06846 [math.NT], 2021.
Gaëtan Chenevier, Publications, in particular The rank n unimodular lattices with no norm 1 vector, for 1<=n<=27 and The rank 28 unimodular lattices with no norm 1 vector.
Gaëtan Chenevier, Unimodular hunting, Modular Forms Workshop, Oberwolfach online, Feb 2021.

Cf. A005134 (cumulative sums), A054908, A054909, A054911.

If 8|n then a(n) = A054908(n) + A054909(n/8), otherwise a(n) = A054908(n). - Andrey Zabolotskiy, Nov 05 2021

a(26)-a(28) added from Gaëtan Chenevier's page by Andrey Zabolotskiy, Nov 05 2021

cp's OEIS Frontend

A005134 Number of n-dimensional unimodular lattices (or quadratic forms).

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A054911 Number of n-dimensional odd unimodular lattices (or quadratic forms).

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A054909 Number of 8n-dimensional even unimodular lattices (or quadratic forms).

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A054907 Number of n-dimensional unimodular lattices (or quadratic forms) containing no vectors of norm 1.

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