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A228884 Determinant of the n X n matrix with (i,j)-entry equal to the greatest common divisor of i-j and n.

%I A228884 #17 Nov 03 2024 19:52:41
%S A228884 1,1,3,20,128,2304,10800,606528,3932160,141087744,1289945088,
%T A228884 210000000000,335544320000,222902511206400,804545281732608,
%U A228884 39137889484800000,972777519512027136,608742554432415203328,391804906912468697088,1455817098785971890290688,968232702940866945220608
%N A228884 Determinant of the n X n matrix with (i,j)-entry equal to the greatest common divisor of i-j and n.
%C A228884 Conjecture: (i) a(n) is always positive and divisible by Phi(n)^{Phi(n)}*sum_{d|n}Phi(d)*n/d, where Phi(n) is Euler's totient function.
%C A228884 (ii) For any composite number n, all prime divisors of a(n) are smaller than n.
%C A228884 It is easy to show that a(n) is divisible by Sum_{d|n} Phi(d)*n/d = Sum_{k=1..n} gcd(k,n), and a(p) = (p-1)^{p-1}*(2p-1) for any prime p.
%H A228884 Alois P. Heinz, <a href="/A228884/b228884.txt">Table of n, a(n) for n = 0..388</a> (terms n = 1..100 from Zhi-Wei Sun)
%e A228884 a(1) = 1 since gcd(1-1,1) = 1.
%p A228884 a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)-> igcd(i-j, n))):
%p A228884 seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 03 2024
%t A228884 a[n_]:=Det[Table[GCD[i-j,n],{i,1,n},{j,1,n}]]
%t A228884 Table[a[n],{n,1,20}]
%Y A228884 Cf. A228885.
%K A228884 nonn
%O A228884 0,3
%A A228884 _Zhi-Wei Sun_, Sep 06 2013
%E A228884 a(0)=1 prepended by _Alois P. Heinz_, Nov 03 2024