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

0,3

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.

Links

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

Example

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

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}]

Crossrefs

Cf. A228885.

Sequence in context: A228750 A187442 A167590 * A138910 A000276 A216778

Adjacent sequences: A228881 A228882 A228883 * A228885 A228886 A228887

Keyword

nonn

Author

Zhi-Wei Sun, Sep 06 2013

Extensions

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

Status

approved