cp's OEIS Frontend

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

%I A380746 #17 May 25 2025 13:22:29
%S A380746 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,
%T A380746 356988
%N A380746 Number of n-dimensional indecomposable unimodular lattices (or quadratic forms).
%C A380746 The sequence {a(n)} is the inverse Euler transform of A005134.
%C A380746 King gives the lower bound a(29) >= 37563933 (using computations of Allombert--Chenevier).
%D A380746 Fu Zu Zhu, Construction of nondecomposable positive definite unimodular quadratic forms. Sci. Sinica Ser. A, 30 (1987), no. 1, 19-31.
%D A380746 Fu Zu Zhu, On nondecomposability and indecomposability of quadratic forms, Sci. Sinica Ser. A, 31 (1988), no. 3, 265-273.
%H A380746 Bill Allombert and Gaëtan Chenevier, <a href="https://arxiv.org/abs/2410.19569">Unimodular Hunting II</a>, arXiv:2410.19569 [math.NT], 2024.
%H A380746 Etsuko Bannai, <a href="https://doi.org/10.1090/memo/0429">Positive definite unimodular lattices with trivial automorphism groups</a>, Mem. Amer. Math. Soc., 85 (1990), no. 429, iv+70 pp.
%H A380746 Gaëtan Chenevier, <a href="https://arxiv.org/abs/2410.18788">Unimodular Hunting</a>, arXiv:2410.18788 [math.NT], 2024.
%H A380746 J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1007/978-1-4757-6568-7">Sphere Packings, Lattices and Groups</a>, Third edition, Springer-Verlag, New York, 1999. lxxiv+703 pp.
%H A380746 Oliver D. King, <a href="https://doi.org/10.1090/S0025-5718-02-01455-2">A mass formula for unimodular lattices with no roots</a>, Math. Comp., 72 (2003), no. 242, 839-863.
%H A380746 O. T. O'Meara, <a href="https://doi.org/10.1515/crll.1975.276.99">The construction of indecomposable positive definite quadratic forms</a>, J. Reine Angew. Math., 276 (1975), 99-123.
%H A380746 Wilhelm Plesken, <a href="https://doi.org/10.1006/jnth.1994.1037">Additively indecomposable positive integral quadratic forms</a>, J. Number Theory, 47 (1994), no. 3, 273-283.
%F A380746 Product_{k>=1} (1-x^k)^(-a(k)) = 1 + Sum_{k>=1} A005134(k)*x^k.
%F A380746 a(n) <= A054907(n) for all n > 1.
%e A380746 For n = 1, the only 1-dimensional indecomposable unimodular lattice is Z, thus a(1) = 1.
%e A380746 For n = 8, the only 8-dimensional indecomposable unimodular lattice is E8, thus a(8) = 1.
%e A380746 For n = 12, the only 12-dimensional indecomposable unimodular lattice is D12+, thus a(12) = 1.
%Y A380746 Cf. A005134, A054907, A054909, A054911, A383643.
%K A380746 nonn,hard,more
%O A380746 1,16
%A A380746 _Robin Visser_, Jan 31 2025