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

1,2

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.

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.

Example

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.

Prog

(Python) # See Branicky link.

Crossrefs

Cf. A000583, A005917.

Sequence in context: A135916 A211812 A266395 * A051880 A007328 A222909

Adjacent sequences: A260547 A260548 A260549 * A260551 A260552 A260553

Keyword

nonn,hard

Author

Aldo González Lorenzo, Jul 29 2015

Status

approved