login
a(n) is the least L^1-norm of a square integer matrix of determinant n. The L^1-norm of the matrix M=(m_i,j) is by definition sum(i,j) |m_i,j|.
0

%I #4 Apr 03 2014 18:44:24

%S 1,2,3,4,5,5,7,6,6,7,9,7,9,9,8,8,10,8,11,9

%N a(n) is the least L^1-norm of a square integer matrix of determinant n. The L^1-norm of the matrix M=(m_i,j) is by definition sum(i,j) |m_i,j|.

%H Daniel Goldstein, Alfred W. Hales, Richard A. Stong, <a href="http://dx.doi.org/10.1090/S0002-9939-2013-11812-5">Light matrices of prime determinant</a>, Proc. Am. Math. Soc. 142 (2014) 805-819

%F It is shown in the paper cited above that lim a(p)/lg(p) = 5/2, where the limit is over primes p tending to infinity and where lg is the logarithm base 2.

%e a(17) = 10 from the 2-by-2 matrix (4 -1\\1 4). This matrix has determinant 17 and L^1-norm 10 = 4 + 1 + 1 + 4. No square integer matrix has determinant 17 and L^1-norm < 10.

%K nonn

%O 1,2

%A Daniel Goldstein (dgoldste(AT)ccrwest.org), Apr 09 2009