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).

1, 2, 13, 271, 12030, 1346758, 214022024, 51763672608, 16088934953136, 6611717516842608, 4412314619046451200, 3533754988232088933120, 3506189715435673999194112, 4444138735439968822425464576, 5893766827264238066914528545792, 8502284313901016361834901076874240, 15350799440394462109333953415858960384

Offset

0,2

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

Links

Table of n, a(n) for n=0..16.

Mathematics Stack Exchange, Determinant of a Toeplitz matrix

Wikipedia, Toeplitz Matrix

Formula

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

Example

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.

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}]]]

Prog

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

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.

Sequence in context: A338424 A134296 A086510 * A334089 A326360 A123113

Adjacent sequences: A356490 A356491 A356492 * A356494 A356495 A356496

Keyword

nonn

Author

Stefano Spezia, Aug 09 2022

Status

approved