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A118772 Determinant of n X n matrix containing the first n^2 3-almost primes in increasing order.

%I A118772 #16 Apr 17 2020 12:16:48
%S A118772 8,-56,156,13328,-920,-83678,1261988,54252742,214409844,-3528354250,
%T A118772 247094703588,-509185323508,15154985424718,884710401396570,
%U A118772 49777180907707320,-172913218088289027,844641410704177098,3066058962037715903,-33948947842497666568
%N A118772 Determinant of n X n matrix containing the first n^2 3-almost primes in increasing order.
%C A118772 3-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 3-almost primes in increasing order, the second column contains the next n 3-almost primes in increasing order, etc. Equivalently, first row contains first n 3-almost primes in increasing order, second row contains next n 3-almost primes in increasing order, etc. See also: A118713 a(n) = semiprime circulant.
%e A118772 a(2) = -56 because of the determinant -56 =
%e A118772 |8, 12|
%e A118772 18, 20|.
%e A118772 a(6) = -83678 because of the determinant -83678 =
%e A118772 | 8, 12, 18, 20, 27, 28|
%e A118772 | 30, 42, 44, 45, 50, 52|
%e A118772 | 63, 66, 68, 70, 75, 76|
%e A118772 | 78, 92, 98, 99, 102, 105|
%e A118772 | 110, 114, 116, 117, 124, 125|
%e A118772 | 130, 138, 147, 148, 153, 154|.
%t A118772 ThreeAlmostPrimePi[ n_ ] := Sum[ PrimePi[ n/(Prime @ i*Prime @ j) ] - j + 1, {i, PrimePi[ n^(1/3) ]}, {j, i, PrimePi@ Sqrt[ n/Prime @ i ]} ]; ThreeAlmostPrime[ n_ ] := Block[ {e = Floor[ Log[ 2, n ] + 1 ], a, b}, a = 2^e; Do[ b = 2^p; While[ ThreeAlmostPrimePi[ a ] < n, a = a + b ]; a = a - b/2, {p, e, 0, -1} ]; a + b/2 ]; Table[ Det[ Partition[ Array[ ThreeAlmostPrime, n^2 ], n ] ], {n, 19} ] (* _Robert G. Wilson v_, May 26 2006 *)
%t A118772 With[{tap=Select[Range[4000],PrimeOmega[#]==3&]},Table[Det[ Partition[ Take[tap,n^2],n]],{n,20}]] (* _Harvey P. Dale_, Apr 17 2020 *)
%Y A118772 Cf. A014612, A067276, A118713, A118770, A118779.
%K A118772 easy,sign
%O A118772 1,1
%A A118772 _Jonathan Vos Post_, May 22 2006
%E A118772 More terms from _Robert G. Wilson v_, May 26 2006
%E A118772 Typos in Mma program corrected by _Giovanni Resta_, Jun 12 2016