A071078 -id:A071078 - cp's OEIS frontend

A071081 Determinant of the n X n matrix whose element (i,j) equals the |i-j|-th composite number, or 0 if i=j.

1, 0, -16, 192, -1904, 16416, -134608, 1102920, -8971103, 69262338, -527129920, 4002967800, -30263030000, 218133853800, -1565386817920, 11130108480678, -75244171093875, 496516351214832, -3261752198331472, 21401161780748720, -140093238345715827, 914525302322457472

Offset: 0

Robert G. Wilson v, May 26 2002

sign

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A071078, A071079, A071080, A071082, A071083.

Cf. A374070 (permanent).

comps:= remove(isprime,[$4 .. 11000]):
f:= proc(n) local M;
  M:= Matrix(n,n,(i,j) -> `if`(i=j,0,comps[abs(i-j)]));
  LinearAlgebra:-Determinant(M)
end proc:
f(0):= 1:
map(f, [$0..25]); # Robert Israel, Dec 02 2024

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; f[n_] := Det[ Table[ If[i == j, 0, Composite[ Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Table[ f[n], {n, 1, 20}]

a(n) = my(composite(n)=my(k=-1); while(-n+n+=-k+k=primepi(n), ); n); matdet(matrix(n, n, i, j, if(i==j, 0, composite(abs(i-j))))); \\ Ruud H.G. van Tol, Jul 14 2024

from sympy import Matrix, composite
def A071081(n): return Matrix(n,n,[composite(abs(j-k)) if j!=k else 0 for j in range(n) for k in range(n)]).det() # Chai Wah Wu, Jul 01 2024

a(21) from Stefano Spezia, Jun 27 2024

a(0)=1 prepended by Alois P. Heinz, Jul 01 2024

A381514 a(n) is the hafnian of a symmetric Toeplitz matrix of order 2*n whose off-diagonal element (i,j) equals the |i-j|-th prime.

Original entry on oeis.org

1, 2, 23, 899, 85072, 15120411, 4439935299, 1989537541918, 1264044973158281, 1090056235155152713, 1227540523199054294506

Offset: 0

Stefano Spezia, Feb 25 2025

			a(2) = 23 because the hafnian of
  [d  2  3  5]
  [2  d  2  3]
  [3  2  d  2]
  [5  3  2  d]
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 2*2 + 3*3 + 5*2 = 23. Here d denotes the generic element on the main diagonal of the matrix from which the hafnian does not depend.

Wikipedia, Hafnian.
Wikipedia, Symmetric matrix.
Wikipedia, Toeplitz Matrix.

Cf. A374067, A374068.

Cf. A071078, A071079, A085807, A306457, A318173, A356483.

M[i_, j_]:=Prime[Abs[i-j]]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i]], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 5, 0]

a(5)-a(10) from Pontus von Brömssen, Feb 26 2025

cp's OEIS Frontend

A071081 Determinant of the n X n matrix whose element (i,j) equals the |i-j|-th composite number, or 0 if i=j.

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