A118983 - OEIS

%I #14 Jan 25 2021 20:36:51

%S 24,12,0,15,30,18,-4,-4,34,-4,-4,22,8,8,0,-8,-8,38,4,4,26,4,4,42,-4,

%T -4,58,-4,-4,50,4,7,-7,-4,52,8,8,0,-8,-8,68,4,4,56,4,4,80,-8,-8,80,4,

%U 4,-4,0,4,-4,-4,86,4,7

%N Determinant of 3 X 3 matrices of n-th continuous block of 9 consecutive composites.

%C Analog of A117330 with composites instead of primes.

%F a(n) = c(n)*c(n+4)*c(n+8) - c(n)*c(n+5)*c(n+7) - c(n+1)*c(n+3)*c(n+8) + c(n+1)*c(n+5)*c(n+6) + c(n+2)*c(n+3)*c(n+7) - c(n+2)*c(n+4)*c(n+6) where c(n) = A002808(n) is the n-th composite.

%e a(1) = 24 =

%e   | 4   6   8|

%e   | 9  10  12|

%e   |14  15  16|.

%e a(3) = 0 because of the first of an infinite number of singular matrices:

%e   | 8   9  10|

%e   |12  14  15|

%e   |16  18  20|.

%e a(15) = 0 because of the singular matrix:

%e   |25  26  27|

%e   |28  30  32|

%e   |33  34  35|.

%e a(38) = 0 because of the singular matrix:

%e   |55  56  57|

%e   |58  60  62|

%e   |63  64  65|.

%e a(54) = 0 because of the singular matrix:

%e   |76  77  78|

%e   |80  81  82|

%e   |84  85  86|.

%p A118983 := proc(n) A002808(n)*A002808(n+4)*A002808(n+8) -A002808(n)*A002808(n+5) *A002808(n+7) -A002808(n+1)*A002808(n+3) *A002808(n+8) +A002808(n+1)*A002808(n+5) *A002808(n+6) +A002808(n+2)*A002808(n+3) *A002808(n+7) -A002808(n+2)*A002808(n+4) *A002808(n+6) ; end proc:

%p seq(A118983(n),n=1..60) ; # _R. J. Mathar_, Dec 22 2010

%t Det[#]&/@(Partition[#,3]&/@Partition[Select[Range[100],CompositeQ],9,1]) (* _Harvey P. Dale_, May 16 2019 *)

%o (PARI) c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808

%o a(n) = matdet(matrix(3,3,i,j,c((n+j-1)+3*(i-1)))); \\ _Michel Marcus_, Jan 25 2021

%Y Cf. A002808, A117301, A117330, A118780, A118781.

%K easy,sign,less

%O 1,1

%A _Jonathan Vos Post_, May 25 2006