A035307 a(n) is the least integer such that every even unimodular lattice in dimension 8n contains some vectors of all even (squared) norm >= 2*a(n).

0, 0, 2, 2, 2, 3

Offset

1,3

Comments

a(4) and a(5) are determined by Odlyzko and Sloane, a(6) by Peters and Kok Seng Chua gives an explicit upper bound for all a(n). Also both a(7) and a(8) are either 2 or 3 as established by Chakraborty et al.

Links

Table of n, a(n) for n=1..6.

K. Chakraborty, A. K. Lal and B. Ramakrishnan, Modular forms that behave like theta series, Math. Computation, Vol. 66, 219, Jul 15 1997, pp. 1169-1183

Kok Seng Chua, An explicit Hecke's bound and exceptions of even unimodular quadratic forms, Bull. Austral. Math. Soc. 65 (2002), 231-238.

A. M. Odlyzko and N. J. A. Sloane, On exceptions of integral quadratic forms, J. reine angew Math. 321, 212-216, (1981)

M. Peters, Definite Unimodular 48-Dimensional Quadratic Forms, Bull. London Math. Soc., 15 (1983), 18-20

Example

a(3)=2 because Leech lattice has no vectors of norm 2. All other 24-dimensional Niemeier lattices contains vectors of all even norms.

Crossrefs

Sequence in context: A239944 A235812 A343471 * A292373 A305383 A004481

Adjacent sequences: A035304 A035305 A035306 * A035308 A035309 A035310

Keyword

nonn,hard,more

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 25 2000

Status

approved