A117330 -id:A117330 - cp's OEIS frontend

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

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

A340923 4*a(n) is the maximum possible determinant of a 3 X 3 matrix whose entries are 9 consecutive primes starting with prime(n).

Original entry on oeis.org

1660, 2693, 3894, 5712, 7030, 9155, 10369, 11718, 14480, 16185, 18774, 20070, 22920, 24720, 23895, 26800, 31560, 39117, 43080, 43245, 42132, 38406, 41056, 48204, 66144, 69006, 86556, 98499, 99021, 88999, 77640, 87348, 86745, 89832, 92466, 95277, 98454, 84820

Offset: 1

Hugo Pfoertner, Jan 26 2021

nonn

The entries of the matrix are arranged in such a way that the determinant of the matrix is maximized.

			a(1) = 1660 = A180128(3)/4 with the corresponding matrix shown in A180128.
a(2) = 2693: determinant (
  [13 29  7]
  [ 3 11 23]
  [19  5 17]) = 10772 = 4*2693.

Cf. A117330, A180128, A340924, A340925.

Table[Max[Det[Partition[#,3]]&/@Permutations[Prime[Range[n,n+8]]]],{n,40}]/4 (* Harvey P. Dale, Jul 21 2021 *)

A340869 Numbers k such that the determinant of the 3 X 3 matrix [prime(k),prime(k+1),prime(k+2); prime(k+3),prime(k+4),prime(k+5); prime(k+6),prime(k+7),prime(k+8)] is a square.

Original entry on oeis.org

4, 12, 14, 131, 222, 229, 330, 351, 356, 525, 561, 825, 969, 979, 1009, 1115, 1123, 1243, 1722, 1826, 2221, 2632, 2673, 2814, 3167, 3436, 3437, 3966, 4056, 4307, 4583, 5010, 5137, 5509, 5772, 6031, 6034, 6230, 6233, 6363, 6505, 6532, 6794, 7112, 7551, 8154, 8330, 8476, 9260, 9348, 9349, 9613

Offset: 1

J. M. Bergot and Robert Israel, Jan 24 2021

nonn

Numbers k such that A117330(k) is a square.

			a(3) = 14 is a term because A117330(14) = Determinant([43,47,53; 59,61,67; 71,73,79]) = 144 = 12^2.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A117330, A340874. Includes A117345.

f:= proc(n) local i; LinearAlgebra:-Determinant(Matrix(3,3,[seq(ithprime(i),i=n..n+8)])) end proc:
select(t -> issqr(f(t)), [$1..10000]);

okQ[k_] := IntegerQ@ Sqrt@ Det@ Partition[Prime[k+#]& /@ Range[0, 8], 3];
Select[Range[10000], okQ] (* Jean-François Alcover, Feb 10 2023 *)

isok(k) = issquare(matdet(matrix(3,3,i,j,prime((k+j-1)+3*(i-1))))); \\ Michel Marcus, Jan 25 2021

from sympy import nextprime, Matrix, integer_nthroot
k,A340869_list, plist = 1,[], [2, 3, 5, 7, 11, 13, 17, 19, 23]
while k < 10**7:
    d = Matrix(plist).reshape(3,3).det()
    if d >= 0 and integer_nthroot(d,2)[1]:
        A340869_list.append(k)
    k,plist = k+1,plist[1:]+[nextprime(plist[-1])] # Chai Wah Wu, Jan 25 2021

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

A340874 Square root of the determinant of the 3 X 3 matrix [prime(k), prime(k+1), prime(k+2); prime(k+3), prime(k+4), prime(k+5); prime(k+6), prime(k+7), prime(k+8)] when that determinant is a square.

Original entry on oeis.org

6, 10, 12, 36, 294, 24, 0, 12, 24, 72, 0, 24, 12, 36, 0, 1564, 0, 12, 12, 0, 156, 0, 12, 60, 36, 48, 24, 0, 0, 72, 60, 60, 24, 60, 12, 0, 12, 12, 12, 0, 0, 12, 180, 0, 60, 0, 60, 72, 120, 0, 120, 0, 2150, 0, 24, 12, 0, 0, 60, 0, 36, 48, 120, 0, 0, 0, 0, 0, 0, 24, 0, 0, 56, 0, 24, 0, 48, 0, 2266

Offset: 1

J. M. Bergot and Robert Israel, Jan 24 2021

nonn

The prime k-tuples conjecture implies that, for example, there are infinitely many k for which the matrix is of the form [x, x+4, x+10; x+22, x+24, x+30; x+34, x+36, x+42], in which case the determinant is 12^2.

			a(3) = 12 because A340869(3) = 14 and the determinant of the 3 X 3 matrix [43, 47, 53; 59, 61, 67; 71, 73, 79] composed of prime(14) to prime(22) in order (by rows or columns) is 144 = 12^2.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A117330, A340869.

f:= proc(n) local i,t;
t:= LinearAlgebra:-Determinant(Matrix(3, 3, [seq(ithprime(i), i=n..n+8)]));
if issqr(t) then sqrt(t) fi
end proc:
map(f, [$1..10000]);

m = 10^4; p = Prime[Range[m + 8]]; Select[Table[Sqrt @ Det @ Partition[p[[n ;; n + 8]], 3], {n, 1, m}], IntegerQ] (* Amiram Eldar, Jan 25 2021 *)

f(n) = matdet(matrix(3,3,i,j,prime((n+j-1)+3*(i-1)))); \\ A117330
lista(nn) = my(x); for (n=1, nn, if (issquare(f(n), &x), print1(x, ", "))); \\ Michel Marcus, Jan 25 2021

a(n) = sqrt(A117330(A340869(n))).

A118983 Determinant of 3 X 3 matrices of n-th continuous block of 9 consecutive composites.

Original entry on oeis.org

24, 12, 0, 15, 30, 18, -4, -4, 34, -4, -4, 22, 8, 8, 0, -8, -8, 38, 4, 4, 26, 4, 4, 42, -4, -4, 58, -4, -4, 50, 4, 7, -7, -4, 52, 8, 8, 0, -8, -8, 68, 4, 4, 56, 4, 4, 80, -8, -8, 80, 4, 4, -4, 0, 4, -4, -4, 86, 4, 7

Offset: 1

Jonathan Vos Post, May 25 2006

Analog of A117330 with composites instead of primes.

			a(1) = 24 =
  | 4   6   8|
  | 9  10  12|
  |14  15  16|.
a(3) = 0 because of the first of an infinite number of singular matrices:
  | 8   9  10|
  |12  14  15|
  |16  18  20|.
a(15) = 0 because of the singular matrix:
  |25  26  27|
  |28  30  32|
  |33  34  35|.
a(38) = 0 because of the singular matrix:
  |55  56  57|
  |58  60  62|
  |63  64  65|.
a(54) = 0 because of the singular matrix:
  |76  77  78|
  |80  81  82|
  |84  85  86|.

Cf. A002808, A117301, A117330, A118780, A118781.

A118983 := proc(n) A002808(n)*A002808(n+4)*A002808(n+8) -A002808(n)*A002808(n+5) *A002808(n+7) -A002808(n+1)*A002808(n+3) *A002808(n+8) +A002808(n+1)*A002808(n+5) *A002808(n+6) +A002808(n+2)*A002808(n+3) *A002808(n+7) -A002808(n+2)*A002808(n+4) *A002808(n+6) ; end proc:
seq(A118983(n),n=1..60) ; # R. J. Mathar, Dec 22 2010

Det[#]&/@(Partition[#,3]&/@Partition[Select[Range[100],CompositeQ],9,1]) (* Harvey P. Dale, May 16 2019 *)

c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = matdet(matrix(3,3,i,j,c((n+j-1)+3*(i-1)))); \\ Michel Marcus, Jan 25 2021

a(n) = c(n)*c(n+4)*c(n+8) - c(n)*c(n+5)*c(n+7) - c(n+1)*c(n+3)*c(n+8) + c(n+1)*c(n+5)*c(n+6) + c(n+2)*c(n+3)*c(n+7) - c(n+2)*c(n+4)*c(n+6) where c(n) = A002808(n) is the n-th composite.

A118875 Determinant of n-th continuous block of 9 consecutive squares of primes.

Original entry on oeis.org

-213720, 114432, -548352, 892800, -1774080, -7289856, 10105344, -79557120, -97790976, 171740160, 147556224, 56531520, -380053440, 122206464, -164292480, -958000320, 394761600, 189907200, 1139760000, -3023127360, -1495428480, -4260988800, -14501393280, 7022695680

Offset: 1

Jonathan Vos Post, May 24 2006

Quadratic analog of A117330 Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive primes. See also: A001248 Squares of primes. The terminology "continuous" is used to distinguish from "discrete" which would be block 1: 4, 9, 25, 49, 121, 169, 289, 361, 529; block 2: 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721; and so forth.

			a(1) = -213720 =
  |  4    9   25|
  | 49  121  169|
  |289  361  529|.
a(2) =
  |   9  25  49|
  | 121 169 289|
  | 361 529 841|.

Cf. A000040, A001248, A117301, A117330.

a:= n-> LinearAlgebra[Determinant](Matrix(3, (i,j)-> ithprime(n+3*i-4+j)^2)):
seq(a(n), n=1..25);  # Alois P. Heinz, Jan 25 2021

m = 24; p = Prime[Range[m + 8]]^2; Table[Det @ Partition[p[[n ;; n + 8]], 3], {n, 1, m}] (* Amiram Eldar, Jan 25 2021 *)

a(n) = matdet(matrix(3, 3, i, j, prime((n+j-1)+3*(i-1))^2)); \\ Michel Marcus, Jan 25 2021

Wrong Formula and data corrected by Michel Marcus, Jan 25 2021

A118876 Determinant of n-th continuous block of 16 consecutive squares of primes.

Original entry on oeis.org

768280320, -1010949120, -4719098880, -1791590400, 24298444800, -19462947840, -109685145600, -3192514560, 144441833472, -198529367040, 15778022400, 159125783040, 861983659008, 1193361776640, 1359501373440, 5328357672960

Offset: 1

Jonathan Vos Post, May 24 2006

Quadratic analog of A118799 Determinants of 4 X 4 matrices of continuous blocks of 16 consecutive primes. See also: A001248 Squares of primes. The terminology "continuous" is used to distinguish from "discrete" which would be block 1: 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809; block 2: 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161; and so forth.

			a(1) = 768280320 =
|...4.....9...25....49.|
|.121...169..289...361.|
|.529...841..961..1369.|
|1681..1849.2209..2809.|.

Cf. A000040, A001248, A117301, A117330, A118799.

A337160 Primes p such that the 3 X 3 matrix with components (row by row) prime(k+m), 0 <= m <= 8 has zero determinant, where p = prime(k).

Original entry on oeis.org

2213, 4073, 8011, 9041, 15649, 23663, 37483, 38453, 59663, 63487, 65111, 71861, 83557, 97157, 100279, 118801, 129527, 131707, 139291, 163601, 166597, 166799, 180181, 180233, 195691, 203807, 209233, 217201, 227561, 238657, 289139, 309121, 327473

Offset: 1

Jianing Song, Jan 28 2021

nonn

Primes arising from A117345.

			The next 8 primes after 2213 are 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, and we have det({{2213, 2221, 2237}, {2239, 2243, 2251}, {2267, 2269, 2273}}) = 0, hence 2213 is a term.

Cf. A117330, A117345, A340923.

for(k=1, 35000, M=matrix(3, 3, i, j, prime(k+3*(i-1)+j-1)); if(matdet(M, 1)==0, print1(prime(k), ", ")))

a(n) = prime(A117345(n)).

cp's OEIS Frontend

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

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A340923 4*a(n) is the maximum possible determinant of a 3 X 3 matrix whose entries are 9 consecutive primes starting with prime(n).

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A340869 Numbers k such that the determinant of the 3 X 3 matrix [prime(k),prime(k+1),prime(k+2); prime(k+3),prime(k+4),prime(k+5); prime(k+6),prime(k+7),prime(k+8)] is a square.

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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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A340874 Square root of the determinant of the 3 X 3 matrix [prime(k), prime(k+1), prime(k+2); prime(k+3), prime(k+4), prime(k+5); prime(k+6), prime(k+7), prime(k+8)] when that determinant is a square.

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A118983 Determinant of 3 X 3 matrices of n-th continuous block of 9 consecutive composites.

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