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A260550
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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.
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1
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1, 15, 75, 237, 559, 1157, 2055, 3471, 5449, 8131, 11633, 16361, 22041, 29349, 38329, 48839, 61325, 76479, 93957, 114717, 138041, 164153, 194505, 229625, 268259, 311031, 359719, 413245, 472145, 537835, 608837, 688121, 774877, 867549, 971403, 1080637, 1198233, 1326059, 1467029, 1617451, 1777881, 1948219, 2132381, 2329081, 2539351
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OFFSET
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1,2
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COMMENTS
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a(n) <= A000583(n), which is the number of 2 X 2 matrices with entries in {1, ..., n}.
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.
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.
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LINKS
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P. F. Rivett and N. I. P. Mackinnon, Prime Matrices, The Mathematical Gazette, Vol. 70, No. 454 (Dec., 1986), pp. 257-259.
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EXAMPLE
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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.
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PROG
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(Python) see link
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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STATUS
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approved
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