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%I #21 Feb 15 2017 21:36:45
%S 0,10,46,114,234,458,826,1370,2090,3010,4174,5658,7534,9930,12954,
%T 16662,21074,26242,32246,39182,47186,56386,66874,78798,92290,107434,
%U 124282,142942,163550,186266,211250,238626,268526,301134,336610,375086,416678,461454,509434,560662,615182,673106
%N Number of 2 X 2 matrices with all elements in 0..n such that the sum of the elements is prime.
%H Charles R Greathouse IV, <a href="/A281550/b281550.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..200 from Indranil Ghosh, terms 201..3000 from Chai Wah Wu)
%e 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.
%e 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.
%o (Python)
%o from sympy import isprime
%o def t(n):
%o ....s=0
%o ....for a in range(0, n+1):
%o ........for b in range(0, n+1):
%o ............for c in range(0, n+1):
%o ................for d in range(0, n+1):
%o ....................if isprime(a+b+c+d)==True:
%o ........................s+=1
%o ....return s
%o for i in range(0, 201):
%o ....print str(i)+" "+str(t(i))
%o (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
%Y Cf. A210000, A281090, A281315.
%K nonn
%O 0,2
%A _Indranil Ghosh_, Jan 23 2017
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