This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383643 #15 May 15 2025 21:28:07 %S A383643 1,0,0,0,0,1,1,1,2 %N A383643 Number of n-dimensional additively indecomposable positive definite integral lattices (or quadratic forms). %C A383643 A positive definite integral lattice (or quadratic form) is additively indecomposable if it cannot be written as a sum of two nonzero positive semidefinite integral lattices (or quadratic forms). Any additively indecomposable lattice is also (orthogonally) indecomposable, although the converse need not hold. %C A383643 By computing all additively indecomposable lattices of determinant up to 100, Opgenorth gave the lower bounds a(10) >= 7 and a(11) >= 13. Eisenbarth showed the lower bounds a(12) >= 29, a(13) >= 9, a(14) >= 10, a(15) >= 9, and a(16) >= 5. %D A383643 Jürgen Opgenorth, Additiv unzerlegbare ganzzahlige quadratische Formen in den Dimensionen 9, 10 und 11. Diplomarbeit, RWTH Aachen, 1992. %H A383643 Simon Eisenbarth, <a href="https://www.math.rwth-aachen.de/homes/Simon.Eisenbarth/bachelor.pdf">Additive Zerlegungen von Gittern</a>, Bachelorarbeit, RWTH Aachen, 2014. %H A383643 Paul Erdős and Chao Ko, <a href="https://doi.org/10.4064/aa-3-1-102-122">On definite quadratic forms, which are not the sum of two definite or semi-definite forms</a>, Acta Arith. 3, 102-122 (1938). %H A383643 Louis J. Mordell, <a href="https://doi.org/10.2307/1968831">The representation of a definite quadratic form as a sum of two others</a>, Ann. of Math. (2) 38 (1937), no. 4, 751-757. %H A383643 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. %H A383643 Ruiqing Wang, <a href="https://doi.org/10.1007/s10114-025-2562-6">Additively indecomposable positive definite integral lattices</a>, Acta Math. Sin. (Engl. Ser.) 41 (2025), no. 3, 908-924. %F A383643 a(n) >= A380746(n). %e A383643 For n <= 8, the only n-dimensional additively indecomposable positive definite lattices are Z (of dimension 1), E6 (of dimension 6), E7 (of dimension 7), and E8 (of dimension 8). %e A383643 For n = 9, the a(9) = 2 additively indecomposable rank 9 positive definite lattices were computed by Opgenorth. These are the two lattices with Gram matrices: %e A383643 [ 2 1 1 1 1 1 1 1 4] [ 2 1 1 1 2 2 2 2 6] %e A383643 [ 1 2 1 1 1 1 1 1 4] [ 1 2 1 1 2 2 2 2 6] %e A383643 [ 1 1 2 1 1 1 1 1 4] [ 1 1 2 1 2 2 2 2 6] %e A383643 [ 1 1 1 2 1 1 1 1 4] [ 1 1 1 2 2 2 2 2 6] %e A383643 [ 1 1 1 1 2 1 1 1 4] [ 2 2 2 2 5 4 4 4 12] %e A383643 [ 1 1 1 1 1 2 1 1 4] [ 2 2 2 2 4 5 4 4 12] %e A383643 [ 1 1 1 1 1 1 2 1 4] [ 2 2 2 2 4 4 5 4 12] %e A383643 [ 1 1 1 1 1 1 1 2 4] [ 2 2 2 2 4 4 4 5 12] %e A383643 [ 4 4 4 4 4 4 4 4 15], [ 6 6 6 6 12 12 12 12 35], %e A383643 having determinant 7 and determinant 15 respectively. %Y A383643 Cf. A380746. %K A383643 nonn,hard,more %O A383643 1,9 %A A383643 _Robin Visser_, May 09 2025