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A340869 Numbers k such that the determinant of the 3 X 3 matrix [prime(k),prime(k+1),prime(k+2); prime(k+3),prime(k+4),prime(k+5); prime(k+6),prime(k+7),prime(k+8)] is a square.

%I A340869 #22 Feb 10 2023 08:08:29
%S A340869 4,12,14,131,222,229,330,351,356,525,561,825,969,979,1009,1115,1123,
%T A340869 1243,1722,1826,2221,2632,2673,2814,3167,3436,3437,3966,4056,4307,
%U A340869 4583,5010,5137,5509,5772,6031,6034,6230,6233,6363,6505,6532,6794,7112,7551,8154,8330,8476,9260,9348,9349,9613
%N A340869 Numbers k such that the determinant of the 3 X 3 matrix [prime(k),prime(k+1),prime(k+2); prime(k+3),prime(k+4),prime(k+5); prime(k+6),prime(k+7),prime(k+8)] is a square.
%C A340869 Numbers k such that A117330(k) is a square.
%H A340869 Robert Israel, <a href="/A340869/b340869.txt">Table of n, a(n) for n = 1..3000</a>
%e A340869 a(3) = 14 is a term because A117330(14) = Determinant([43,47,53; 59,61,67; 71,73,79]) = 144 = 12^2.
%p A340869 f:= proc(n) local i; LinearAlgebra:-Determinant(Matrix(3,3,[seq(ithprime(i),i=n..n+8)])) end proc:
%p A340869 select(t -> issqr(f(t)), [$1..10000]);
%t A340869 okQ[k_] := IntegerQ@ Sqrt@ Det@ Partition[Prime[k+#]& /@ Range[0, 8], 3];
%t A340869 Select[Range[10000], okQ] (* _Jean-François Alcover_, Feb 10 2023 *)
%o A340869 (PARI) isok(k) = issquare(matdet(matrix(3,3,i,j,prime((k+j-1)+3*(i-1))))); \\ _Michel Marcus_, Jan 25 2021
%o A340869 (Python)
%o A340869 from sympy import nextprime, Matrix, integer_nthroot
%o A340869 k,A340869_list, plist = 1,[], [2, 3, 5, 7, 11, 13, 17, 19, 23]
%o A340869 while k < 10**7:
%o A340869     d = Matrix(plist).reshape(3,3).det()
%o A340869     if d >= 0 and integer_nthroot(d,2)[1]:
%o A340869         A340869_list.append(k)
%o A340869     k,plist = k+1,plist[1:]+[nextprime(plist[-1])] # _Chai Wah Wu_, Jan 25 2021
%Y A340869 Cf. A117330, A340874.  Includes A117345.
%K A340869 nonn
%O A340869 1,1
%A A340869 _J. M. Bergot_ and _Robert Israel_, Jan 24 2021