A380746 Number of n-dimensional indecomposable unimodular lattices (or quadratic forms).

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 3, 11, 12, 27, 48, 176, 367, 1896, 14489, 356988

Offset: 1

Author

Robin Visser, Jan 31 2025

Keywords

• nonn
• hard
• more

Comments

The sequence {a(n)} is the inverse Euler transform of A005134.

King gives the lower bound a(29) >= 37563933 (using computations of Allombert--Chenevier).

Examples

			For n = 1, the only 1-dimensional indecomposable unimodular lattice is Z, thus a(1) = 1.
For n = 8, the only 8-dimensional indecomposable unimodular lattice is E8, thus a(8) = 1.
For n = 12, the only 12-dimensional indecomposable unimodular lattice is D12+, thus a(12) = 1.
		

References

• Fu Zu Zhu, Construction of nondecomposable positive definite unimodular quadratic forms. Sci. Sinica Ser. A, 30 (1987), no. 1, 19-31.
• Fu Zu Zhu, On nondecomposability and indecomposability of quadratic forms, Sci. Sinica Ser. A, 31 (1988), no. 3, 265-273.

Links

• Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
• Etsuko Bannai, Positive definite unimodular lattices with trivial automorphism groups, Mem. Amer. Math. Soc., 85 (1990), no. 429, iv+70 pp.
• Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Third edition, Springer-Verlag, New York, 1999. lxxiv+703 pp.
• Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863.
• O. T. O'Meara, The construction of indecomposable positive definite quadratic forms, J. Reine Angew. Math., 276 (1975), 99-123.
• Wilhelm Plesken, Additively indecomposable positive integral quadratic forms, J. Number Theory, 47 (1994), no. 3, 273-283.

Crossrefs

Cf. A005134, A054907, A054909, A054911, A383643.

Formula

Product_{k>=1} (1-x^k)^(-a(k)) = 1 + Sum_{k>=1} A005134(k)*x^k.

a(n) <= A054907(n) for all n > 1.