A118983 Determinant of 3 X 3 matrices of n-th continuous block of 9 consecutive composites.

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, -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, 4, -4, 0, 4, -4, -4, 86, 4, 7

Offset

1,1

Comments

Analog of A117330 with composites instead of primes.

Links

Table of n, a(n) for n=1..60.

Formula

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.

Example

a(1) = 24 =
  | 4   6   8|
  | 9  10  12|
  |14  15  16|.
a(3) = 0 because of the first of an infinite number of singular matrices:
  | 8   9  10|
  |12  14  15|
  |16  18  20|.
a(15) = 0 because of the singular matrix:
  |25  26  27|
  |28  30  32|
  |33  34  35|.
a(38) = 0 because of the singular matrix:
  |55  56  57|
  |58  60  62|
  |63  64  65|.
a(54) = 0 because of the singular matrix:
  |76  77  78|
  |80  81  82|
  |84  85  86|.

Maple

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:
seq(A118983(n), n=1..60) ; # R. J. Mathar, Dec 22 2010

Mathematica

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

Prog

(PARI) c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = matdet(matrix(3, 3, i, j, c((n+j-1)+3*(i-1)))); \\ Michel Marcus, Jan 25 2021

Crossrefs

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

Sequence in context: A343389 A267024 A259337 * A033968 A033344 A228437

Adjacent sequences: A118980 A118981 A118982 * A118984 A118985 A118986

Keyword

easy,sign,less

Author

Jonathan Vos Post, May 25 2006

Status

approved