A281315 Number of 2 X 2 matrices with all elements in {0,...,n} and prime determinant.

0, 0, 13, 46, 83, 191, 272, 509, 687, 1010, 1291, 2019, 2364, 3468, 4132, 5079, 6072, 8298, 9234, 12189, 13621, 15984, 18095, 22965, 24886, 29942, 33248, 38385, 42073, 51053, 53882, 64609, 70619, 78663, 85424, 96024, 101521, 118804, 127940, 140598, 149375, 172123, 179424, 205334, 218216

Offset: 0

Author

Indranil Ghosh, Jan 20 2017

Keywords

• nonn

Examples

			For n = 3, a few of the possible matrices are [1,0;3,3], [1,1;0,2], [1,1;0,3], [1,1;1,3], [1,2;0,2], [1,2;0,3], [1,3;0,2], [1,3;0,3], [2,0;0,1], [2,0;1,1], [2,0;2,1], [2,0;3,1], [2,1;0,1], [2,1;1,2], [2,1;1,3], [3,1;3,2], [3,2;0,1], [3,2;1,3], [3,2;2,2], [3,2;2,3], ... There are 46 possibilities.
Here each of the matrices M is defined as M = [a,b;c,d], where a= M[1][1], b = M[1][2], c = M[2][1] and d = M[2][2]. So, a(3) = 46.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..1000 (terms 0..186 from Indranil Ghosh)

Crossrefs

Cf. A210000.

Programs

Python

from sympy import isprime
def t(n):
    s=0
    for a in range(n+1):
        for d in range(n+1):
            ad = a * d
            for c in range(n+1):
                for b in range(n+1):
                    if isprime(ad-b*c):
                        s+=1
    return s
for i in range(187):
    print(str(i)+" "+str(t(i)))

Sage

def A281315(n):
    T = Tuples([i for i in range(n+1)], 4); i = 0
    for t in T: i += is_prime(t[0]*t[3]-t[1]*t[2])
    return i
[A281315(n) for n in range(20)] # Peter Luschny, Jul 23 2017