Le Paige's Theorem
Let 
be the 
matrix whose 
th entry is 1 if 
divides 
and 0 otherwise, let 
be the 
diagonal matrix 
, where 
is the totient function, and let 
be the 
matrix whose 
th entry is the greatest common divisor 
. Then Le Paige's theorem states that
where 
denotes the transpose (Le Paige 1878, Johnson 2003).
As a corollary,
(Smith 1876, Johnson 2003). For 
, 2, ... the first few values are 1, 1, 2, 4, 16, 32, 192, 768, ... (OEIS A001088).
REFERENCES:
Johnson, W. P. "An 
Factorization in Elementary Number Theory." Math. Mag. 76, 392-394, 2003.
Le Paige, C. "Sur un théorème de M. Mansion." Nouv. Corresp. Math. 4, 176-178, 1878.
Mansion, P. "On an Arithmetical Theorem of Professor Smith's." Messenger Math. 7, 81-82, 1877.
Muir, T. A Treatise on the Theory of Determinants, Vol. 3. New York: Dover, 1960.
Sloane, N. J. A. Sequence A001088 in "The On-Line Encyclopedia of Integer Sequences."
Smith, H. J. S. "On the Value of a Certain Arithmetical Determinant." Proc. London Math. Soc. 7, 208-212, 1876. Reprinted in The Collected Mathematical Papers of Henry John Stephen Smith, Vol. 2 (Ed. J. W. L. Glaisher). Oxford, England: Clarendon Press, pp. 161-165, 1894.
