What the least common multiple divides
Let \(a_ 1, a_ 2,\dots\) be a sequence satisfying \((a_ m, a_ n)= a_{(m,n)}\) for all \(m\), \(n\), where \((\cdot, \cdot)\) stands for the greatest common divisor. Let \(L(n)\) be the l.c.m., and \(P(n)\) the product of the first \(n\) terms of the sequence. The main result of the paper is that \(L(n)\) divides the quotient of \(P(n)\) by \(P(n/2) P(n/3) P(n/7) P(n/43)\dots\), where each term in the sequence \(2,3,7,43,\dots\) is one greater than the product of all the preceding terms. The extent to which this result is best possible is also discussed in the paper.
scientific article; zbMATH DE number 638547