A067276
Determinant of n X n matrix containing the first n^2 primes in increasing order.
Original entry on oeis.org
2, -1, -78, 880, -4656, -14304, -423936, 8342720, 711956736, -615707136, 21057138688, -4663930678272, 211912980656128, -9178450735677440, 40005919124799488, 83013253447139328, -8525111273818357760, -800258888289188708352, -15170733077495639179264
Offset: 1
a(3) = -78 because det[[2,7,17],[3,11,19],[5,13,23]] = -78 (= det[[2,3,5],[7,11,13],[17,19,23]], the determinant of the transpose.).
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[ Determinant( Matrix(n, n, [ NthPrime(k): k in [1..n^2] ]) ): n in [1..19] ]; // Klaus Brockhaus, May 12 2010
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seq(LinearAlgebra:-Determinant(Matrix(n,n,(i,j) -> ithprime(n*(i-1)+j))),n=1..20); # Robert Israel, Jul 12 2017
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Table[ Det[ Partition[ Array[Prime, n^2], n]], {n, 19}] (* Robert G. Wilson v, May 26 2006 *)
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for(n=1,20,k=0; m=matrix(n,n,x,y, prime(k=k+1)); print1(matdet(m), ", ")) /* The matrix initialization command above fills columns first: Variables (such as) x and y take on values 1 through n for rows and columns, respectively, with x changing more rapidly and they must be specified even though the 5th argument is not an explicit function of them here. */
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from sympy.matrices import Matrix
from sympy import sieve
def a(n):
sieve.extend_to_no(n**2)
return Matrix(n, n, sieve[1:n**2+1]).det()
print([a(n) for n in range(1, 20)]) # Indranil Ghosh, Jul 31 2017
A118770
Determinant of n X n matrix containing the first n^2 semiprimes in increasing order.
Original entry on oeis.org
4, -14, -196, 480, 696, -57901, -525364, -409579, 18528507, -237549252, -2119519900, 6713972874, 18262155072, -19072020914992, 162234208372185, 1471912942112734, 6828673030820538, -35126752028893500, 729026655790306778, -15365360727898374618
Offset: 1
a(2) = -14 because of the determinant -14 =
|4,6 |
|9,10|.
a(6) = -57901 = the determinant
|4, 6, 9, 10, 14, 15,|
|21, 22, 25, 26, 33, 34,|
|35, 38, 39, 46, 49, 51,|
|55, 57, 58, 62, 65, 69,|
|74, 77, 82, 85, 86, 87,|
|91, 93, 94, 95, 106, 111|.
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SemiPrimePi[ n_ ] := Sum[ PrimePi[ n/Prime @ i ] - i + 1, {i, PrimePi @ Sqrt @ n} ]; SemiPrime[ n_ ] := Block[ {e = Floor[ Log[ 2, n ] + 1 ], a, b}, a = 2^e; Do[ b = 2^p; While[ SemiPrimePi[ a ] < n, a = a + b ]; a = a - b/2, {p, e, 0, -1} ]; a + b/2 ]; Table[ Det[ Partition[ Array[ SemiPrime, n^2 ], n ] ], {n, 20} ] (* Robert G. Wilson v, May 26 2006 *)
Module[{nn=5000,spr},spr=Select[Range[nn],PrimeOmega[#]==2&];Table[Det[ Partition[ Take[spr,n^2],n]],{n,Sqrt[Length[spr]]}]] (* Harvey P. Dale, Nov 21 2018 *)
A118772
Determinant of n X n matrix containing the first n^2 3-almost primes in increasing order.
Original entry on oeis.org
8, -56, 156, 13328, -920, -83678, 1261988, 54252742, 214409844, -3528354250, 247094703588, -509185323508, 15154985424718, 884710401396570, 49777180907707320, -172913218088289027, 844641410704177098, 3066058962037715903, -33948947842497666568
Offset: 1
a(2) = -56 because of the determinant -56 =
|8, 12|
18, 20|.
a(6) = -83678 because of the determinant -83678 =
| 8, 12, 18, 20, 27, 28|
| 30, 42, 44, 45, 50, 52|
| 63, 66, 68, 70, 75, 76|
| 78, 92, 98, 99, 102, 105|
| 110, 114, 116, 117, 124, 125|
| 130, 138, 147, 148, 153, 154|.
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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 *)
With[{tap=Select[Range[4000],PrimeOmega[#]==3&]},Table[Det[ Partition[ Take[tap,n^2],n]],{n,20}]] (* Harvey P. Dale, Apr 17 2020 *)
Showing 1-3 of 3 results.
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