cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A351021 Minimal permanent of an n X n symmetric Toeplitz matrix using the first n prime numbers.

Original entry on oeis.org

1, 2, 13, 166, 4009, 169469, 10949857, 1078348288, 138679521597, 24402542896843, 5348124003487173
Offset: 0

Views

Author

Stefano Spezia, Jan 29 2022

Keywords

Examples

			a(3) = 166:
    3    2    5
    2    3    2
    5    2    3
a(4) = 4009:
    3    2    5    7
    2    3    2    5
    5    2    3    2
    7    5    2    3
a(5) = 169469:
    5    2    3    7   11
    2    5    2    3    7
    3    2    5    2    3
    7    3    2    5    2
   11    7    3    2    5

Links

Crossrefs

Cf. A348891, A350939, A350955, A351022 (maximal).

Programs

  • Python
    from itertools import permutations
from sympy import Matrix, prime
def A351021(n): return 1 if n == 0 else min(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

Extensions

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

A369833 a(n) is the number of distinct values of the permanent of an n X n symmetric Toeplitz matrix using the first n prime numbers.

Original entry on oeis.org

1, 1, 1, 6, 24, 120, 720, 5040, 40320, 362880
Offset: 0

Views

Author

Stefano Spezia, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A000142, A351021, A351022.
Cf. A369830, A369831, A369832, A369834, A369835.

Programs

  • Mathematica
    a[n_] := CountDistinct[Table[Permanent[ToeplitzMatrix[Part[Permutations[Prime[Range[n]]], i]]], {i, n !}]]; Join[{1}, Array[a,9]]
  • Python
    from itertools import permutations
from sympy import primerange, prime, Matrix
def A369833(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).per() for p in permutations(primerange(prime(n)+1))}) if n else 1 # Chai Wah Wu, Feb 11 2024

Formula

a(n) <= A000142(n).
Conjectured e.g.f.: 1/(1 - x) - x^2/2.

A356491 a(n) is the permanent 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, 13, 184, 4745, 215442, 14998965, 1522204560, 208682406913, 37467772675962, 8809394996942597, 2597094620811897948, 954601857873086235553, 428809643170145564168434, 229499307540038336275308821, 144367721963876506217872778284, 106064861375232790889279725340713
Offset: 0

Views

Author

Stefano Spezia, Aug 09 2022

Keywords

Comments

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

Examples

			For n = 1 the matrix M(1) is
    2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
    2, 3
    3, 2
with permanent a(2) = 13.
For n = 3 the matrix M(3) is
    2, 3, 5
    3, 2, 3
    5, 3, 2
with permanent a(3) = 184.

Links

Crossrefs

Cf. A005843 (trace of the matrix M(n)), A309131 (k-superdiagonal sum of the matrix M(n)), A356483 (hafnian of the matrix M(2*n)), A356490 (determinant of the matrix M(n)).
Cf. A351021, A351022.

Programs

  • Maple
    A356491 := proc(n)
    local c ;
    if n =0 then
        return 1 ;
    end if;
    LinearAlgebra[ToeplitzMatrix]([seq(ithprime(c),c=1..n)],n,symmetric) ;
    LinearAlgebra[Permanent](%) ;
end proc:
seq(A356491(n),n=0..15) ; # R. J. Mathar, Jan 31 2023
  • Mathematica
    k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Array[k, n]]; a[n_]:=Permanent[M[n]]; Join[{1},Table[a[n],{n,16}]]
  • PARI
    a(n) = matpermanent(apply(prime, matrix(n,n,i,j,abs(i-j)+1))); \\ Michel Marcus, Aug 12 2022
  • Python
    from sympy import Matrix, prime
def A356491(n): return Matrix(n,n,[prime(abs(i-j)+1) for i in range(n) for j in range(n)]).per() if n else 1 # Chai Wah Wu, Aug 12 2022

Formula

A351021(n) <= a(n) <= A351022(n).

A351611 Maximal permanent of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Original entry on oeis.org

1, 1, 11, 420, 41451, 7985639, 2779152652
Offset: 0

Views

Author

Stefano Spezia, Feb 14 2022

Keywords

Examples

			a(3) = 420:
    1    5    6
    5    3    4
    6    4    2
a(4) = 41451:
    1    5    8   10
    5    4    9    7
    8    9    3    6
   10    7    6    2

Crossrefs

Cf. A000217, A351153, A351610 (minimal).
Cf. A350566, A350859, A350938, A350940, A351020, A351022.

Extensions

a(5)-a(6) from Hugo Pfoertner, Feb 15 2022

A356493 a(n) is the permanent 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, 13, 271, 12030, 1346758, 214022024, 51763672608, 16088934953136, 6611717516842608, 4412314619046451200, 3533754988232088933120, 3506189715435673999194112, 4444138735439968822425464576, 5893766827264238066914528545792, 8502284313901016361834901076874240, 15350799440394462109333953415858960384
Offset: 0

Views

Author

Stefano Spezia, Aug 09 2022

Keywords

Comments

Conjecture: a(n) is prime only for n = 1, 2, and 3.
Conjecture is true because a(n) is even for n >= 4. This is because a(n) == A356492(n) (mod 2), and all but two rows of the matrix consist of odd numbers. - Robert Israel, Oct 13 2023

Examples

			For n = 1 the matrix M(1) is
    2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
    3, 2
    2, 3
with permanent a(2) = 13.
For n = 3 the matrix M(3) is
    5, 3, 2
    3, 5, 3
    2, 3, 5
with permanent a(3) = 271.

Links

Crossrefs

Cf. A033286 (trace of the matrix M(n)), A356484 (hafnian of the matrix M(2*n)), A356492 (determinant of the matrix M(n)).
Cf. A351021, A351022.

Programs

  • Mathematica
    k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Reverse[Array[k, n]]]; a[n_]:=Permanent[M[n]]; PrimeQ[Join[{1},Table[a[n],{n,16}]]]
  • PARI
    a(n) = matpermanent(apply(prime, matrix(n,n,i,j,n-abs(i-j)))); \\ Michel Marcus, Aug 12 2022

Formula

A351021(n) <= a(n) <= A351022(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

Views

Author

Stefano Spezia, Jul 14 2023

Keywords

Examples

			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;
  ...

Links

Crossrefs

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).

Programs

  • Mathematica
    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
  • PARI
    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

Extensions

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

Views

Author

Stefano Spezia, Jul 14 2023

Keywords

Links

Crossrefs

Right diagonal of A364233.
Cf. A348891 (minimal nonzero absolute value determinant), A350955 (minimal determinant), A350956 (maximal determinant), A351021 (minimal permanent), A351022 (maximal permanent).

Programs

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

Extensions

a(10)-a(12) from Andrew Howroyd, Dec 31 2023
Showing 1-7 of 7 results.

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