A215914 Smallest positive integer k such that there is no k-dimensional unital *-subalgebra of the n X n complex matrices.

2, 3, 4, 7, 8, 13, 16, 23, 24, 43, 44, 49, 64, 77, 80, 97, 116, 141, 144, 167, 168, 193, 248, 249, 280, 313, 348, 385, 424, 473, 484, 527, 528, 573, 620, 625, 720, 725, 828, 833, 890, 949, 1010, 1073, 1088, 1153, 1220, 1289, 1360, 1433

Offset

1,1

Comments

a(n) is the smallest positive integer that is not contained in the n-th row of A215905.

a(n) >= n+1. In fact, for any m >= 1 there exists N >= 1 such that a(n) > mn for all n >= N. That is, this sequence grows super-linearly.

Links

Table of n, a(n) for n=1..50.

Example

In the n = 4 case, there are unital *-subalgebras of dimensions 1 though 6, as follows:
a 0 0 0 ... a 0 0 0 ... a 0 0 0 ... a 0 0 0 ... a 0 0 0 ... a 0 0 0
0 a 0 0 ... 0 b 0 0 ... 0 b 0 0 ... 0 b 0 0 ... 0 a 0 0 ... 0 b 0 0
0 0 a 0 ... 0 0 b 0 ... 0 0 c 0 ... 0 0 c 0 ... 0 0 b c ... 0 0 c d
0 0 0 a ... 0 0 0 b ... 0 0 0 c ... 0 0 0 d ... 0 0 d e ... 0 0 e f
However, there is no unital *-subalgebra of the 4-by-4 matrices of dimension 7, so a(4) = 7.

Crossrefs

Cf. A215905, A215909.

Sequence in context: A370121 A057887 A202116 * A006049 A084541 A240690

Adjacent sequences: A215911 A215912 A215913 * A215915 A215916 A215917

Keyword

nonn

Author

Nathaniel Johnston, Aug 26 2012

Status

approved