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

1, 1, 3, 20, 128, 2304, 10800, 606528, 3932160, 141087744, 1289945088, 210000000000, 335544320000, 222902511206400, 804545281732608, 39137889484800000, 972777519512027136, 608742554432415203328, 391804906912468697088, 1455817098785971890290688, 968232702940866945220608

Offset: 0

Author

Zhi-Wei Sun, Sep 06 2013

Keywords

• nonn

Comments

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.

(ii) For any composite number n, all prime divisors of a(n) are smaller than n.

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.

Examples

			a(1) = 1 since gcd(1-1,1) = 1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..388 (terms n = 1..100 from Zhi-Wei Sun)

Crossrefs

Cf. A228885.

Programs

Maple

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)-> igcd(i-j, n))):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 03 2024

Mathematica

a[n_]:=Det[Table[GCD[i-j,n],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,20}]

Extensions

a(0)=1 prepended by Alois P. Heinz, Nov 03 2024