A118713 -id:A118713 - cp's OEIS frontend

A118770 Determinant of n X n matrix containing the first n^2 semiprimes in increasing order.

4, -14, -196, 480, 696, -57901, -525364, -409579, 18528507, -237549252, -2119519900, 6713972874, 18262155072, -19072020914992, 162234208372185, 1471912942112734, 6828673030820538, -35126752028893500, 729026655790306778, -15365360727898374618

Offset: 1

Jonathan Vos Post, May 22 2006

Semiprime 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 semiprimes in increasing order, the second column contains the next n semiprimes in increasing order, etc. Equivalently, first row contains first n semiprimes in increasing order, second row contains next n semiprimes in increasing order, etc. See also: A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.

			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|.

Harvey P. Dale, Table of n, a(n) for n = 1..570

Cf. A001358, A067276, A118713, A118772, A118779.

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 *)

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

Typos in Mma program corrected by Giovanni Resta, Jun 12 2016

A118799 Determinants of 4 X 4 matrices of 16 consecutive primes.

Original entry on oeis.org

880, -448, -1472, -240, 2480, -1352, -4128, -96, 2736, -2520, 120, 1080, 4288, 4880, 4600, 13368, 7056, 14560, 2960, 13320, 0, 24864, -11096, -24264, 0, -9168, -2128, -15792, 0, 18120, -5248, 6384, -21840, -38776, -20480, 20176, -72896, -69200, 40080, -37632

Offset: 1

Jonathan Vos Post, May 23 2006

4 X 4 analog of A117330.

All terms are even. - Harvey P. Dale, May 05 2016

			a(1) = 880 =
  | 2  3  5  7|
  |11 13 17 19|
  |23 29 31 37|
  |41 43 47 53|.
a(10) = -2520 =
  |29 31 37 41|
  |43 47 53 59|
  |61 67 71 73|
  |79 83 89 97|.
a(21) = 0 =
  | 73  79  83  89|
  | 97 101 103 107|
  |109 113 127 131|
  |137 139 149 151|.

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

Cf. A000040, A067276, A117301, A117330 , A118713.

A118799 := proc(n)
    local A,i,r,c ;
    A := Matrix(4,4) ;
    i := n ;
    for r from 1 to 4 do
    for c from 1 to 4 do
        A[r,c] := ithprime(i) ;
        i := i+1 ;
    end do:
    end do:
    LinearAlgebra[Determinant](A) ;
end proc: # R. J. Mathar, May 05 2013

Module[{nn=60,prs},prs=Prime[Range[nn]];Table[Det[Partition[ Take[ prs, {n,n+15}],4]],{n,nn-15}]] (* Harvey P. Dale, Apr 29 2016 *)

a(n) = matdet(matrix(4,4,i,j,prime((n+j-1)+4*(i-1)))); \\ Michel Marcus, Jan 25 2021

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

Jonathan Vos Post, May 22 2006

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.

			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|.

Cf. A014612, A067276, A118713, A118770, A118779.

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 *)

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

Typos in Mma program corrected by Giovanni Resta, Jun 12 2016

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

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, May 22 2006

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.

			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|.

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

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 *)

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

A118780 Semiprime(n)semiprime(n+3) - semiprime(n+1)semiprime(n+2), where semiprime(n) is the n-th semiprime.

Original entry on oeis.org

-14, -6, -5, 0, -7, -87, -4, 76, -8, -212, 64, -4, 128, 68, -265, 31, -12, -177, 104, 109, -28, 103, -101, -40, -24, -348, -176, 253, 81, -285, -97, 928, 364, -841, -257, -361, -127, -3, -125, 603, 359, -675, 367, -8, -860, 139, -3, 995, 280, -1276, -167, 629, 145, 443, -365, -579, 171, -569

Offset: 1

Jonathan Vos Post, May 22 2006

Semiprime analog of A117301.

By construction, every entry is also the difference between two 4-almost primes: a(1) = A014613(4)-A014613(5); a(2) = A014613(9)-A014613(11); a(3) = A014613(16)-A014613(18); a(4) = A014613(27)-A014613(27); etc. - R. J. Mathar, Nov 27 2007

			a(1) = -14 because the determinant of the first block of 4 consecutive semiprimes is:
|4. 6.|
|9. 10|.
a(4) = 0 because the determinant of the 4th block of 4 semiprimes is the first of a presumably infinite number of singular matrices:
|10. 14.|
|15. 21.|.
a(8) = 76, the first positive value in the sequence:
|22. 25.|
|26. 33.|.

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

Cf. A001358, A067276, A117301, A118713.

A001358 := proc(n) option remember ; local a; if n = 1 then 4 ; else for a from A001358(n-1)+1 do if numtheory[bigomega](a)= 2 then RETURN(a) ; fi ; od: fi ; end: A118780 := proc(n) A001358(n)*A001358(n+3)-A001358(n+1)*A001358(n+2) ; end: seq(A118780(n),n=1..58) ; # R. J. Mathar, Nov 27 2007

nmax = 58; spmax = nmax; SP = {};
While[nmax+3 > Length[SP], spmax += nmax; SP = Select[Range[spmax], PrimeOmega[#] == 2&]];
a[n_] := SP[[n]] SP[[n+3]] - SP[[n+1]] SP[[n+2]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Aug 01 2023 *)
#[[1]]#[[4]]-#[[2]]#[[3]]&/@Partition[Select[Range[300],PrimeOmega[#]==2&],4,1] (* Harvey P. Dale, Sep 08 2024 *)

a(n) = A001358(n)*A001358(n+3) - A001358(n+1)*A001358(n+2).

Better definition from Jens Kruse Andersen, May 03 2008

A118781 Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive semiprimes.

Original entry on oeis.org

-196, 272, -251, 149, -423, 909, -408, -452, 958, -123, -112, -460, 84, -271, -187, -162, 63, 7, 101, -483, -133, 205, -860, -46, 339, 1178, 848, 366, 1084, 719, -384, 334, -2736, -984, -1912, 214, 34, 40, -1735, -60, 64, -2263, -3468, 5795, 69, 132, 3007, 256, 2130, 3428

Offset: 1

Jonathan Vos Post, May 22 2006

Semiprime analog of A117330 Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive primes. The terminology "continuous" is used to distinguish from "discrete" which would be (in this 3 X 3 semiprime case) block 1: 4, 6, 9, 10, 14, 15, 21, 22, 25; block 2: 26, 33, 34, 35, 38, 39, 46, 49, 51; and so forth.

			a(1) = -196 because the determinant of the first continuous block of 9 semiprimes is:
| 4. 6. 9.|
|10. 14. 15.|
|21. 22. 25.|.
a(9) = 958 because the determinant of the 9th continuous block of 9 semiprimes is:
|25. 26. 33.|
|34. 35. 38.|
|39. 46. 49.|.
a(50) = 3428 because the determinant of the 50th continuous block of 9 semiprimes is:
|146. 155. 158.|
|159. 161. 166.|
|169. 177. 178.|.

Cf. A001358, A067276, A117301, A118713.

A118781 := proc(n)
    local A,i,r,c ;
    A := Matrix(3,3) ;
    i := n ;
    for r from 1 to 3 do
    for c from 1 to 3 do
        A[r,c] := A001358(i) ;
        i := i+1 ;
    end do:
    end do:
    LinearAlgebra[Determinant](A) ;
end proc: # R. J. Mathar, May 05 2013

Det/@(Partition[#,3]&/@(Partition[Select[Range[200],PrimeOmega[ #] == 2&],9,1])) (* Harvey P. Dale, Nov 29 2015 *)

a(n) = s(n)*s(n+4)*s(n+8) - s(n)*s(n+5)*s(n+7) - s(n+1)*s(n+3)*s(n+8) + s(n+1)*s(n+5)*s(n+6) + s(n+2)*s(n+3)*s(n+7) - s(n+2)*s(n+4)*s(n+6) where s(n) = A001358(n) is the n-th semiprime.

A118815 Determinants of 5 X 5 matrices consisting of 25 consecutive primes.

Original entry on oeis.org

-4656, 1440, 2912, 2832, -10464, -768, -17376, 20384, -72976, -18944, 12672, 41184, -199776, 28944, -21104, 3552, 25488, 338448, -60192, 39952, -21792, -161904, 499488, -83424, -7440, 7440, -54288, -75456, 1641792, 42288

Offset: 1

Jonathan Vos Post, May 23 2006

5 X 5 analog of A117330.

			a(1) = -4656 =
  | 2  3  5  7 11|
  |13 17 19 23 29|
  |31 37 41 43 47|
  |53 59 61 67 71|
  |73 79 83 89 97|.

Cf. A000040, A067276, A117301, A117330, A118713.

a(n) = matdet(matrix(5,5,i,j,prime((n+j-1)+5*(i-1)))); \\ Michel Marcus, Jan 25 2021

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A118770 Determinant of n X n matrix containing the first n^2 semiprimes in increasing order.

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A118799 Determinants of 4 X 4 matrices of 16 consecutive primes.

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

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

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A118780 Semiprime(n)*semiprime(n+3) - semiprime(n+1)*semiprime(n+2), where semiprime(n) is the n-th semiprime.

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A118781 Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive semiprimes.

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A118815 Determinants of 5 X 5 matrices consisting of 25 consecutive primes.

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A118780 Semiprime(n)semiprime(n+3) - semiprime(n+1)semiprime(n+2), where semiprime(n) is the n-th semiprime.