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a(n) is the number of 2 X 2 matrices with entries in {1, ..., n} that are not the product of two 2 X 2 positive integer matrices.
1

%I #32 Jan 01 2025 22:20:50

%S 1,15,75,237,559,1157,2055,3471,5449,8131,11633,16361,22041,29349,

%T 38329,48839,61325,76479,93957,114717,138041,164153,194505,229625,

%U 268259,311031,359719,413245,472145,537835,608837,688121,774877,867549,971403,1080637,1198233,1326059,1467029,1617451,1777881,1948219,2132381,2329081,2539351

%N a(n) is the number of 2 X 2 matrices with entries in {1, ..., n} that are not the product of two 2 X 2 positive integer matrices.

%C a(n) <= A000583(n), which is the number of 2 X 2 matrices with entries in {1, ..., n}.

%C a(n) >= A005917(n), which is the number of 2 X 2 matrices with entries in {1, ..., n} that contain the element 1. All such matrices are not decomposable as a product of 2 X 2 positive integer matrices.

%C This definition is a generalization of the notion of prime numbers to the family of 2 X 2 positive integer matrices. Since the matrices do not contain 0, max(A*B) > max(A) and max(A*B) > max(B). Thus, for every matrix there is a finite number of possible decompositions to check.

%H Michael S. Branicky, <a href="/A260550/b260550.txt">Table of n, a(n) for n = 1..60</a>

%H Michael S. Branicky, <a href="/A260550/a260550.py.txt">Python program</a>

%H Aldo González Lorenzo, <a href="/A260550/a260550.txt">Scilab function for computing this sequence</a>

%H P. F. Rivett and N. I. P. Mackinnon, <a href="http://www.jstor.org/stable/3616179">Prime Matrices</a>, The Mathematical Gazette, Vol. 70, No. 454 (Dec., 1986), pp. 257-259.

%e The matrix [2,2;3,3] is decomposable: [2,2;3,3] = [1,1;1,2] * [1,1;1,1]. However, the matrix [2,3;3;2] is not decomposable.

%o (Python) # See Branicky link.

%Y Cf. A000583, A005917.

%K nonn,hard

%O 1,2

%A _Aldo González Lorenzo_, Jul 29 2015