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%I #10 Sep 07 2013 09:47:06
%S 1,3,20,128,2304,10800,606528,3932160,141087744,1289945088,
%T 210000000000,335544320000,222902511206400,804545281732608,
%U 39137889484800000,972777519512027136,608742554432415203328,391804906912468697088,1455817098785971890290688,968232702940866945220608
%N Determinant of the n X n matrix with (i,j)-entry equal to the greatest common divisor of i-j and n.
%C 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 (ii) For any composite number n, all prime divisors of a(n) are smaller than n.
%C 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 Zhi-Wei Sun, <a href="/A228884/b228884.txt">Table of n, a(n) for n = 1..100</a>
%e a(1) = 1 since gcd(1-1,1) = 1.
%t a[n_]:=Det[Table[GCD[i-j,n],{i,1,n},{j,1,n}]]
%t Table[a[n],{n,1,20}]
%Y Cf. A228885.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Sep 06 2013
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