A281550 Number of 2 X 2 matrices with all elements in 0..n such that the sum of the elements is prime.
0, 10, 46, 114, 234, 458, 826, 1370, 2090, 3010, 4174, 5658, 7534, 9930, 12954, 16662, 21074, 26242, 32246, 39182, 47186, 56386, 66874, 78798, 92290, 107434, 124282, 142942, 163550, 186266, 211250, 238626, 268526, 301134, 336610, 375086, 416678, 461454, 509434, 560662, 615182, 673106
Offset: 0
Keywords
Examples
For n = 4, a few of the possible matrices are [0,4;2,1], [0,4;3,0], [0,4;3,4], [0,4;4,3], [1,0;0,1], [1,0;0,2], [1,0;0,4], [1,0;1,0], [1,0;1,1], [1,0;1,3], [2,2;3,0], [2,2;3,4], [2,2;4,3], [2,3;0,0], [2,3;0,2], [3,4;3,3], [3,4;4,0], [3,4;4,2], [4,0;0,1], [4,0;0,3], [4,0;1,0], ... There are 234 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], d = M[2][2]. So, a(4) = 234.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms 0..200 from Indranil Ghosh, terms 201..3000 from Chai Wah Wu)
Programs
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PARI
a(n)=my(X=Pol(vector(n+1,i,1))+O('x^(4*n)),Y=X^4,s); forprime(p=2,4*n, s+=polcoeff(Y,p)); s \\ Charles R Greathouse IV, Feb 15 2017 -
Python
from sympy import isprime def t(n): s=0 for a in range(0, n+1): for b in range(0, n+1): for c in range(0, n+1): for d in range(0, n+1): if isprime(a+b+c+d)==True: s+=1 return s for i in range(0, 201): print(str(i)+" "+str(t(i)))
Formula
a(n) = Sum_{p prime} Sum_{k=0..4} (-1)^k * binomial(4, k) * binomial(p+3-k*(n+1), 3). - David Radcliffe, Jun 13 2025