

A215914


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


3



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
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OFFSET

1,1


COMMENTS

a(n) is the smallest positive integer that is not contained in the nth 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 superlinearly.


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 4by4 matrices of dimension 7, so a(4) = 7.


CROSSREFS

Cf. A215905, A215909.
Sequence in context: A066847 A057887 A202116 * A006049 A084541 A240690
Adjacent sequences: A215911 A215912 A215913 * A215915 A215916 A215917


KEYWORD

nonn


AUTHOR

Nathaniel Johnston, Aug 26 2012


STATUS

approved



