A350956 -id:A350956 - cp's OEIS frontend

A350955 Minimal determinant of an n X n symmetric Toeplitz matrix using the first n prime numbers.

1, 2, -5, -35, -435, -87986, -7186995, -496722800, -68316404507, -9102428703537, -3721326642272925, -488684390484513105, -195315251884652232704

Offset: 0

Stefano Spezia, Jan 27 2022

			a(3) = -35:
   [3    5    2]
   [5    3    5]
   [2    5    3]
a(4) = -435:
   [5    7    2    3]
   [7    5    7    2]
   [2    7    5    7]
   [3    2    7    5]
a(5) = -87986:
   [ 2    3   11    5    7]
   [ 3    2    3   11    5]
   [11    3    2    3   11]
   [ 5   11    3    2    3]
   [ 7    5   11    3    2]

Lucas A. Brown, A348891+A350955+6.py
Wikipedia, Toeplitz Matrix

Cf. A350932, A350956 (maximal), A348891.

from itertools import permutations
from sympy import Matrix, prime
def A350955(n): return min(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).det() for p in permutations(prime(i) for i in range(1,n+1))) # Chai Wah Wu, Jan 27 2022

a(9) from Alois P. Heinz, Jan 27 2022

a(10)-a(12) from Lucas A. Brown, Aug 29 2022

A351022 Maximal permanent of an n X n symmetric Toeplitz matrix using the first n prime numbers.

Original entry on oeis.org

1, 2, 13, 289, 13814, 1795898, 265709592, 70163924440, 20610999526800, 9097511018219760, 6845834489829830144

Offset: 0

Stefano Spezia, Jan 29 2022

			a(3) = 289:
    3    5    2
    5    3    5
    2    5    3
a(4) = 13814:
    5    7    3    2
    7    5    7    3
    3    7    5    7
    2    3    7    5
a(5) = 1795898:
    5   11    7    3    2
   11    5   11    7    3
    7   11    5   11    7
    3    7   11    5   11
    2    3    7   11    5

Lucas A. Brown, A351021+2.sage
Wikipedia, Toeplitz Matrix

Cf. A350940, A350956, A351021 (minimal).

from itertools import permutations
from sympy import Matrix, prime
def A351022(n): return 1 if n == 0 else max(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).per() for p in permutations(prime(i) for i in range(1,n+1))) # Chai Wah Wu, Jan 31 2022

a(9) and a(10) from Lucas A. Brown, Sep 04 2022

A369832 a(n) is the number of distinct values of the determinant of an n X n symmetric Toeplitz matrix using the first n prime numbers.

Original entry on oeis.org

1, 1, 2, 6, 24, 116, 717, 5033, 40301, 362845

Offset: 0

Stefano Spezia, Feb 03 2024

Wikipedia, Toeplitz Matrix.

Cf. A000142, A348891, A350955, A350956.

Cf. A369830, A369831, A369833, A369834, A369835.

a[n_] := CountDistinct[Table[Det[ToeplitzMatrix[Part[Permutations[Prime[Range[n]]], i]]], {i, n !}]]; Join[{1}, Array[a,9]]

from itertools import permutations
from sympy import primerange, prime, Matrix
def A369832(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).det() for p in permutations(primerange(prime(n)+1))}) if n else 1 # Chai Wah Wu, Feb 11 2024

a(n) <= A000142(n).

A351609 Maximal absolute value of the determinant of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Original entry on oeis.org

1, 1, 7, 152, 7113, 745285, 94974369

Offset: 0

Stefano Spezia, Feb 14 2022

Upper bounds for the next terms can be found by considering all possibilities of choosing matrix entries on the diagonal and applying Gasper's determinant theorem (see references in A085000): a(7) <= 22475584128, a(8) <= 6634478203404, a(9) <= 2647044512044258. - Hugo Pfoertner, Feb 18 2022

			a(3) = 152:
   2    4    6
   4    5    1
   6    1    3
a(4) = 7113:
   2    6    8    9
   6    5   10    1
   8   10    3    4
   9    1    4    7

Cf. A000217, A351147, A351148, A351153.

Cf. A085000, A180128, A350931, A350933, A350954, A350956.

a(n) = max(abs(A351147(n)), A351148(n)). - Hugo Pfoertner, Feb 16 2022

a(5)-a(6) from Hugo Pfoertner, Feb 16 2022

A356490 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).

Original entry on oeis.org

1, 2, -5, 12, -19, -22, 1143, -4284, 14265, -46726, -84405, 1306096, 32312445, 522174906, 4105967871, 5135940112, -642055973735, -2832632334858, 14310549077571, 380891148658140, 4888186898996125, -49513565563840210, 383405170118692791, -2517836083641473036, -3043377347606882055

Offset: 0

Stefano Spezia, Aug 09 2022

sign

Conjecture: abs(a(n)) is prime only for n = 1, 2, and 4.

			For n = 1 the matrix M(1) is
    2
with determinant a(1) = 2.
For n = 2 the matrix M(2) is
    2, 3
    3, 2
with determinant a(2) = -5.
For n = 3 the matrix M(3) is
    2, 3, 5
    3, 2, 3
    5, 3, 2
with determinant a(3) = 12.

Mathematics Stack Exchange, Determinant of a Toeplitz matrix
Wikipedia, Toeplitz Matrix

Cf. A005843 (trace of M(n)), A309131 (k-superdiagonal sum of M(n)), A356483 (hafnian of M(2*n)), A356491 (permanent of M(n)).

Cf. A350955, A350956.

A356490 := proc(n)
    local T,c ;
    if n =0 then
        return 1 ;
    end if;
    T := LinearAlgebra[ToeplitzMatrix]([seq(ithprime(c),c=1..n)],n,symmetric) ;
    LinearAlgebra[Determinant](T) ;
end proc:
seq(A356490(n),n=0..15) ; # R. J. Mathar, Jan 31 2023

k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Array[k, n]]; a[n_]:=Det[M[n]]; Join[{1},Table[a[n],{n,24}]]

a(n) = matdet(apply(prime, matrix(n,n,i,j,abs(i-j)+1))); \\ Michel Marcus, Aug 12 2022

from sympy import Matrix, prime
def A356490(n): return Matrix(n,n,[prime(abs(i-j)+1) for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Aug 12 2022

A350955(n) <= a(n) <= A350956(n).

A356492 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(n), prime(n-1), ..., prime(1).

Original entry on oeis.org

1, 2, 5, 51, 264, 19532, -11904, 1261296, -2052864, 70621632, 24618221568, 3996020736, 743171562496, 24567175118848, -1257930752000, 864893030400, 12289833785344000, 1099483729459478528, 100515455071223808, 757166323365314560, 6294658173770137600, 7801939905505132544

Offset: 0

Stefano Spezia, Aug 09 2022

sign

Conjecture: a(n) is prime only for n = 1 and 2.

Conjecture is true because a(n) is even for n >= 4. This is because all but two rows of the matrix consist of odd numbers. - Robert Israel, Oct 13 2023

			For n = 1 the matrix M(1) is
    2
with determinant a(1) = 2.
For n = 2 the matrix M(2) is
    3, 2
    2, 3
with determinant a(2) = 5.
For n = 3 the matrix M(3) is
    5, 3, 2
    3, 5, 3
    2, 3, 5
with determinant a(3) = 51.

Robert Israel, Table of n, a(n) for n = 0..532
Mathematics Stack Exchange, Determinant of a Toeplitz matrix
Wikipedia, Toeplitz Matrix

Cf. A033286 (trace of the matrix M(n)), A356484 (hafnian of the matrix M(2*n)), A356493 (permanent of the matrix M(n)).

Cf. A350955, A350956.

f:=proc(n) uses LinearAlgebra; local i;
 Determinant(ToeplitzMatrix([seq(ithprime(i),i=n..1,-1)],symmetric));
end proc:
q(0):= 1:
map(q, [$0..25]); # Robert Israel, Oct 13 2023

k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Reverse[Array[k, n]]]; a[n_]:=Det[M[n]]; Join[{1},Table[a[n],{n,21}]]

a(n) = matdet(apply(prime, matrix(n,n,i,j,n-abs(i-j)))); \\ Michel Marcus, Aug 12 2022

A350955(n) <= a(n) <= A350956(n).

A364233 Triangle read by rows: T(n, k) is the number of n X n symmetric Toeplitz matrices of rank k using all the first n prime numbers integers.

Original entry on oeis.org

1, 0, 2, 0, 0, 6, 0, 0, 0, 24, 0, 0, 0, 0, 120, 0, 0, 0, 0, 2, 718, 0, 0, 0, 0, 0, 4, 5036, 0, 0, 0, 0, 0, 1, 3, 40316, 0, 0, 0, 0, 0, 0, 0, 18, 362862, 0, 0, 0, 0, 0, 0, 0, 0, 14, 3628786, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, 39916701, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 78, 479001517

Offset: 1

Stefano Spezia, Jul 14 2023

			The triangle begins:
  1;
  0, 2;
  0, 0, 6;
  0, 0, 0, 24;
  0, 0, 0,  0, 120;
  0, 0, 0,  0,   2, 718;
  0, 0, 0,  0,   0,   4, 5036;
  ...

Wikipedia, Toeplitz Matrix

Cf. A000142 (row sums), A348891 (minimal nonzero absolute value determinant), A350955 (minimal determinant), A350956 (maximal determinant), A351021 (minimal permanent), A351022 (maximal permanent), A364234 (right diagonal).

T[n_,k_]:= Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Prime[Range[n]]], i]]],{i,n!}],k]; Table[T[n,k],{n,8},{k,n}]//Flatten

MkMat(v)={matrix(#v, #v, i, j, v[1+abs(i-j)])}
row(n)={my(f=vector(n)); forperm(vector(n,i,prime(i)), v, f[matrank(MkMat(v))]++); f} \\ Andrew Howroyd, Dec 31 2023

Terms a(46) and beyond from Andrew Howroyd, Dec 31 2023

A364234 a(n) is the number of n X n nonsingular symmetric Toeplitz matrices using all the first n prime numbers.

Original entry on oeis.org

1, 2, 6, 24, 120, 718, 5036, 40316, 362862, 3628786, 39916701, 479001517

Offset: 1

Stefano Spezia, Jul 14 2023

Wikipedia, Toeplitz Matrix

Right diagonal of A364233.

Cf. A348891 (minimal nonzero absolute value determinant), A350955 (minimal determinant), A350956 (maximal determinant), A351021 (minimal permanent), A351022 (maximal permanent).

a[n_]:=Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Prime[Range[n]]], i]]], {i, n!}], n]; Array[a, 8]

a(10)-a(12) from Andrew Howroyd, Dec 31 2023

cp's OEIS Frontend

A350955 Minimal determinant of an n X n symmetric Toeplitz matrix using the first n prime numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

A351022 Maximal permanent of an n X n symmetric Toeplitz matrix using the first n prime numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

A369832 a(n) is the number of distinct values of the determinant of an n X n symmetric Toeplitz matrix using the first n prime numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A351609 Maximal absolute value of the determinant of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A356490 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A356492 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(n), prime(n-1), ..., prime(1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A364233 Triangle read by rows: T(n, k) is the number of n X n symmetric Toeplitz matrices of rank k using all the first n prime numbers integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A364234 a(n) is the number of n X n nonsingular symmetric Toeplitz matrices using all the first n prime numbers.

Views