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.