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

%I A118779 #7 Jul 11 2015 11:15:06
%S A118779 16,-224,0,182016,12734992,-80430368,-125120640,1334967760,
%T A118779 1060202222660,-2759409121760,54820105989504,-14148083510835712,
%U A118779 49989643415528010,299304923505836144,1713123391839442498,93227182153040103540,-86403659709730762670
%N A118779 Determinant of n X n matrix containing the first n^2 4-almost primes in increasing order.
%C A118779 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.
%e A118779 a(2) = -224 because of the determinant -224 =
%e A118779 |16, 24|
%e A118779 |36, 40|.
%e A118779 a(3) = 0 because this matrix is singular: 0 =
%e A118779 |16, 24, 36|
%e A118779 |40, 54, 56|
%e A118779 |60, 81, 84|.
%e A118779 a(6) = -80430368 because of the determinant -80430368 =
%e A118779 | 16, 24, 36, 40, 54, 56|
%e A118779 | 60, 81, 84, 88, 90, 100|
%e A118779 | 104, 126, 132, 135, 136, 140|
%e A118779 | 150, 152, 156, 184, 189, 196|
%e A118779 | 198, 204, 210, 220, 225, 228|
%e A118779 | 232, 234, 248, 250, 260, 276|.
%e A118779 a(8) = 1334967760 =
%e A118779 | 16, 24, 36, 40, 54, 56, 60, 81|
%e A118779 | 84, 88, 90, 100, 104, 126, 132, 135|
%e A118779 |136, 140, 150, 152, 156, 184, 189, 196|
%e A118779 |198, 204, 210, 220, 225, 228, 232, 234|
%e A118779 |248, 250, 260, 276, 294, 296, 297, 306|
%e A118779 |308, 315, 328, 330, 340, 342, 344, 348|
%e A118779 |350, 351, 364, 372, 375, 376, 380, 390|
%e A118779 |414, 424, 441, 444, 459, 460, 462, 472|.
%t A118779 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 *)
%Y A118779 Cf. A014613, A118713, A067276, A118770, A118772.
%K A118779 easy,sign
%O A118779 1,1
%A A118779 _Jonathan Vos Post_, May 22 2006
%E A118779 More terms from _Robert G. Wilson v_, May 26 2006