A118780 -id:A118780 - cp's OEIS frontend

A118877 Determinant of n-th continuous block of 4 consecutive composites.

-12, -12, 6, 6, -18, -18, 12, 12, -24, -24, 18, -3, -28, -2, -2, 24, 24, -36, -36, -2, -2, 32, -3, -42, 36, 36, -48, -48, 42, -3, -52, -2, -2, 48, -3, -58, -2, -2, 54, 54, -66, -66, -2, -2, 62, -3, -72, 66, 66, -78, -78, -2, -2, 74, -3, -84, 78, -3, -88, -2, -2

Offset: 1

Jonathan Vos Post, May 24 2006

Composites analog of A117301 Determinants of 2 X 2 matrices of continuous blocks of 4 consecutive primes. See also: A118780 Determinants of 2 X 2 matrices of continuous blocks of 4 consecutive semiprimes. The terminology "continuous" is used to distinguish from "discrete" which would be (in this composites case) block 1: 4, 6, 8, 9; block 2: 10, 12, 14, 15 and so forth. It is not until a(12) that we break the pattern of a(2n)=a(2n-1); there seem to be strangely many such pairs of two identical values. a(12) is also the first odd value in the sequence and the first prime.

			a(1) = -12 =
|4 6|
|8 9|.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002808, A117301, A118780.

Module[{nn=100,cmps},cmps=Select[Range[nn],CompositeQ];Det[ ArrayReshape[ #,{2,2}]]&/@Table[Take[cmps,{n,n+3}],{n,Length[cmps]-3}]] (* Harvey P. Dale, Aug 03 2020 *)

a(n) = A002808(n)*A002808(n+3) - A002808(n+1)*A002808(n+2).

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

Original entry on oeis.org

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

Jonathan Vos Post, May 25 2006

Analog of A117330 with composites instead of primes.

			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|.

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

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

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

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

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.

cp's OEIS Frontend

A118877 Determinant of n-th continuous block of 4 consecutive composites.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula