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%I #8 Dec 16 2016 02:46:35
%S 2,3,4,7,8,13,16,23,24,43,44,49,64,77,80,97,116,141,144,167,168,193,
%T 248,249,280,313,348,385,424,473,484,527,528,573,620,625,720,725,828,
%U 833,890,949,1010,1073,1088,1153,1220,1289,1360,1433
%N Smallest positive integer k such that there is no k-dimensional unital *-subalgebra of the n X n complex matrices.
%C a(n) is the smallest positive integer that is not contained in the n-th row of A215905.
%C 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.
%e In the n = 4 case, there are unital *-subalgebras of dimensions 1 though 6, as follows:
%e a 0 0 0 ... a 0 0 0 ... a 0 0 0 ... a 0 0 0 ... a 0 0 0 ... a 0 0 0
%e 0 a 0 0 ... 0 b 0 0 ... 0 b 0 0 ... 0 b 0 0 ... 0 a 0 0 ... 0 b 0 0
%e 0 0 a 0 ... 0 0 b 0 ... 0 0 c 0 ... 0 0 c 0 ... 0 0 b c ... 0 0 c d
%e 0 0 0 a ... 0 0 0 b ... 0 0 0 c ... 0 0 0 d ... 0 0 d e ... 0 0 e f
%e However, there is no unital *-subalgebra of the 4-by-4 matrices of dimension 7, so a(4) = 7.
%Y Cf. A215905, A215909.
%K nonn
%O 1,1
%A _Nathaniel Johnston_, Aug 26 2012
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