A260550 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, 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

Offset: 1

Author

Aldo González Lorenzo, Jul 29 2015

Keywords

• nonn
• hard

Comments

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.

Examples

			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.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..60
• Michael S. Branicky, Python program
• Aldo González Lorenzo, Scilab function for computing this sequence
• P. F. Rivett and N. I. P. Mackinnon, Prime Matrices, The Mathematical Gazette, Vol. 70, No. 454 (Dec., 1986), pp. 257-259.

Crossrefs

Cf. A000583, A005917.

Programs

Python

# See Branicky link.