A309839 a(n) = GAP_n: first integer m that is not the dimension of a semisimple subalgebra of M_n(k).

3, 6, 7, 12, 15, 22, 23, 42, 43, 48, 63, 76, 79, 96, 115, 140, 143, 166, 167, 192, 247, 248, 279, 312, 347, 384, 423, 472, 483, 526, 527, 572, 619, 624, 719, 724, 827, 832, 889, 948, 1009, 1072, 1087, 1152, 1219, 1288, 1359, 1432, 1507, 1520, 1597, 1676, 1679

Offset: 2

Author

Phillip Heikoop, Aug 19 2019

Keywords

• nonn

Comments

Define the sequence a(n) = GAP_n to be the smallest integer that is not the dimension of a semisimple subalgebra of M_n(k). This is one more than the upper endpoint of the continuous region of M_n(k). Because when n = 1 there are no gaps, this sequence begins at n = 2. See Heikoop paper, page 31.

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..101
• Phillip Tomas Heikoop, Dimensions of Matrix Subalgebras, Bachelor's Thesis, Worcester Polytechnic Institute (2019).
• Phillip Heikoop, C++11 code to generate the sequence

Crossrefs

Cf. A000124, A002620, A069999, A138544, A309838.

Formula

a(n) > n^2 - 4 * sqrt(n + 2).