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Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive semiprimes.
2

%I #11 Nov 29 2015 13:44:29

%S -196,272,-251,149,-423,909,-408,-452,958,-123,-112,-460,84,-271,-187,

%T -162,63,7,101,-483,-133,205,-860,-46,339,1178,848,366,1084,719,-384,

%U 334,-2736,-984,-1912,214,34,40,-1735,-60,64,-2263,-3468,5795,69,132,3007,256,2130,3428

%N Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive semiprimes.

%C Semiprime analog of A117330 Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive primes. The terminology "continuous" is used to distinguish from "discrete" which would be (in this 3 X 3 semiprime case) block 1: 4, 6, 9, 10, 14, 15, 21, 22, 25; block 2: 26, 33, 34, 35, 38, 39, 46, 49, 51; and so forth.

%F a(n) = s(n)*s(n+4)*s(n+8) - s(n)*s(n+5)*s(n+7) - s(n+1)*s(n+3)*s(n+8) + s(n+1)*s(n+5)*s(n+6) + s(n+2)*s(n+3)*s(n+7) - s(n+2)*s(n+4)*s(n+6) where s(n) = A001358(n) is the n-th semiprime.

%e a(1) = -196 because the determinant of the first continuous block of 9 semiprimes is:

%e | 4. 6. 9.|

%e |10. 14. 15.|

%e |21. 22. 25.|.

%e a(9) = 958 because the determinant of the 9th continuous block of 9 semiprimes is:

%e |25. 26. 33.|

%e |34. 35. 38.|

%e |39. 46. 49.|.

%e a(50) = 3428 because the determinant of the 50th continuous block of 9 semiprimes is:

%e |146. 155. 158.|

%e |159. 161. 166.|

%e |169. 177. 178.|.

%p A118781 := proc(n)

%p local A,i,r,c ;

%p A := Matrix(3,3) ;

%p i := n ;

%p for r from 1 to 3 do

%p for c from 1 to 3 do

%p A[r,c] := A001358(i) ;

%p i := i+1 ;

%p end do:

%p end do:

%p LinearAlgebra[Determinant](A) ;

%p end proc: # _R. J. Mathar_, May 05 2013

%t Det/@(Partition[#,3]&/@(Partition[Select[Range[200],PrimeOmega[ #] == 2&],9,1])) (* _Harvey P. Dale_, Nov 29 2015 *)

%Y Cf. A001358, A067276, A117301, A118713.

%K easy,sign

%O 1,1

%A _Jonathan Vos Post_, May 22 2006