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

%I A071081 #25 Dec 03 2024 03:27:40
%S A071081 1,0,-16,192,-1904,16416,-134608,1102920,-8971103,69262338,-527129920,
%T A071081 4002967800,-30263030000,218133853800,-1565386817920,11130108480678,
%U A071081 -75244171093875,496516351214832,-3261752198331472,21401161780748720,-140093238345715827,914525302322457472
%N A071081 Determinant of the n X n matrix whose element (i,j) equals the |i-j|-th composite number, or 0 if i=j.
%H A071081 Robert Israel, <a href="/A071081/b071081.txt">Table of n, a(n) for n = 0..1000</a>
%p A071081 comps:= remove(isprime,[$4 .. 11000]):
%p A071081 f:= proc(n) local M;
%p A071081   M:= Matrix(n,n,(i,j) -> `if`(i=j,0,comps[abs(i-j)]));
%p A071081   LinearAlgebra:-Determinant(M)
%p A071081 end proc:
%p A071081 f(0):= 1:
%p A071081 map(f, [$0..25]); # _Robert Israel_, Dec 02 2024
%t A071081 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}]
%o A071081 (Python)
%o A071081 from sympy import Matrix, composite
%o A071081 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
%o A071081 (PARI) 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
%Y A071081 Cf. A071078, A071079, A071080, A071082, A071083.
%Y A071081 Cf. A374070 (permanent).
%K A071081 sign
%O A071081 0,3
%A A071081 _Robert G. Wilson v_, May 26 2002
%E A071081 a(21) from _Stefano Spezia_, Jun 27 2024
%E A071081 a(0)=1 prepended by _Alois P. Heinz_, Jul 01 2024