A118779 Determinant of n X n matrix containing the first n^2 4-almost primes in increasing order.

16, -224, 0, 182016, 12734992, -80430368, -125120640, 1334967760, 1060202222660, -2759409121760, 54820105989504, -14148083510835712, 49989643415528010, 299304923505836144, 1713123391839442498, 93227182153040103540, -86403659709730762670

Offset: 1

Author

Jonathan Vos Post, May 22 2006

Keywords

• easy
• sign

Comments

4-almost prime analog of A067276 Determinant of n X n matrix containing the first n^2 primes in increasing order. The first column contains the first n 4-almost primes (A014613) in increasing order, the second column contains the next n 4-almost primes in increasing order, etc. Equivalently, first row contains first n 4-almost primes in increasing order, second row contains next n 4-almost primes in increasing order, etc. See also: A118713 a(n) = semiprime circulant.

Examples

			a(2) = -224 because of the determinant -224 =
|16, 24|
|36, 40|.
a(3) = 0 because this matrix is singular: 0 =
|16, 24, 36|
|40, 54, 56|
|60, 81, 84|.
a(6) = -80430368 because of the determinant -80430368 =
| 16, 24, 36, 40, 54, 56|
| 60, 81, 84, 88, 90, 100|
| 104, 126, 132, 135, 136, 140|
| 150, 152, 156, 184, 189, 196|
| 198, 204, 210, 220, 225, 228|
| 232, 234, 248, 250, 260, 276|.
a(8) = 1334967760 =
| 16, 24, 36, 40, 54, 56, 60, 81|
| 84, 88, 90, 100, 104, 126, 132, 135|
|136, 140, 150, 152, 156, 184, 189, 196|
|198, 204, 210, 220, 225, 228, 232, 234|
|248, 250, 260, 276, 294, 296, 297, 306|
|308, 315, 328, 330, 340, 342, 344, 348|
|350, 351, 364, 372, 375, 376, 380, 390|
|414, 424, 441, 444, 459, 460, 462, 472|.
		

Crossrefs

Cf. A014613, A118713, A067276, A118770, A118772.

Programs

Mathematica

FourAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}]; FourAlmostPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[FourAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Table[Det[Partition[Array[FourAlmostPrime, n^2], n]], {n, 17}] (* Robert G. Wilson v, May 26 2006 *)

Extensions

More terms from Robert G. Wilson v, May 26 2006