Phillip Heikoop - cp's OEIS frontend

User: Phillip Heikoop

Phillip Heikoop has authored 2 sequences.

A309838 a(n) = number of dimensions of semisimple matrix subalgebras.

2, 4, 7, 11, 16, 22, 29, 39, 50, 60, 73, 88, 103, 120, 139, 160, 181, 203, 229, 256, 284, 313, 343, 377, 412, 448, 487, 528, 569, 610, 653, 699, 748, 797, 849, 904, 959, 1014, 1070, 1129, 1191, 1255, 1321, 1388, 1456, 1526, 1598, 1672, 1746, 1821, 1899, 1981, 2064

Offset: 1

Phillip Heikoop, Aug 19 2019

nonn

a(n) = |A_n| in the Heikoop paper.

A semisimple matrix subalgebra of M_n(k) for an algebraically closed field k is a direct sum of M_n_i(k) such that Sum (n_i) <= n. See Heikoop paper, Section 3.2, for more.

Phillip Heikoop, Table of n, a(n) for n = 1..336
Phillip Tomas Heikoop, Dimensions of Matrix Subalgebras, Bachelor's Thesis, Worcester Polytechnic Institute (2019).
Phillip Heikoop, C++11 code to generate the sequence

Cf. A000124, A002620, A069999, A138544, A309839.

a(n) <= n^2 - Sqrt(2) * Sqrt(2n+ 3) * n.

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

Original entry on oeis.org

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

Phillip Heikoop, Aug 19 2019

nonn

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.

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

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

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

cp's OEIS Frontend

User: Phillip Heikoop

A309838 a(n) = number of dimensions of semisimple matrix subalgebras.

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