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%I #9 Mar 01 2020 12:46:46
%S 0,0,2,2,2,3
%N 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).
%C 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.
%H K. Chakraborty, A. K. Lal and B. Ramakrishnan, <a href="https://doi.org/10.1090/S0025-5718-97-00872-7">Modular forms that behave like theta series</a>, Math. Computation, Vol. 66, 219, Jul 15 1997, pp. 1169-1183
%H Kok Seng Chua, <a href="https://doi.org/10.1017/S000497270002027X">An explicit Hecke's bound and exceptions of even unimodular quadratic forms</a>, Bull. Austral. Math. Soc. 65 (2002), 231-238.
%H A. M. Odlyzko and N. J. A. Sloane, <a href="https://www.researchgate.net/publication/240146256_On_exceptions_of_integral_quadratic_forms">On exceptions of integral quadratic forms</a>, J. reine angew Math. 321, 212-216, (1981)
%H M. Peters, <a href="https://doi.org/10.1112/blms/15.1.18">Definite Unimodular 48-Dimensional Quadratic Forms</a>, Bull. London Math. Soc., 15 (1983), 18-20
%e a(3)=2 because Leech lattice has no vectors of norm 2. All other 24-dimensional Niemeier lattices contains vectors of all even norms.
%K nonn,hard,more
%O 1,3
%A Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 25 2000