My question is essentially the $m=2$ case of this question.

Given a positive integer $k$, I'm interested in (small) pairs of positive integers $a,b$ such that every positive integer up to (and including) $k$ is a factor of at least one of them. For example, for $k=10$, one such pair is $a=70$, $b=72$; every integer up to 10 is a factor of at least one of these two numbers.

It's easy to see that the product of any such pair must be a multiple of the least common multiple, call it $L(k)$, of $1,2,3,\dots,k$. It's known that $L(k)$ is asymptotic to $e^k$. Moreover, it's trivial to find a pair whose product is exactly $L(k)$; just take $a=1$, $b=L(k)$. This tells me that, for this problem, the product, $ab$, is not a good measure of how small the pair $a,b$ is. Two other measures that suggest themselves are the sum, $a+b$, and the maximum. Since the product is at least $L(k)$, the sum must be at least $2\sqrt{L(k)}$, and the maximum must be at least $\sqrt{L(k)}$. My question is, how sharp are these bounds?

I did a small amount of calculation by hand during a recent committee meeting, and arrived at these figures, given without any guarantee that they are, in fact, minimal: $$\matrix{k&a&b&a+b&ab/L\cr3&2&3&5&1\cr4&3&4&7&1\cr5&5&12&17&1\cr6&5&12&17&1\cr7&12&35&47&1\cr7&28&30&58&2\cr8&24&35&59&1\cr9&35&72&107&1\cr10&70&72&142&2\cr11&77&360&437&1\cr12&77&360&437&1\cr13&360&1001&1361&1\cr}$$ Note that for $k=7$ I have given two $a,b$ pairs, one with a smaller sum, the other with a smaller maximum.

I have checked the Online Encyclopedia of Integer Sequences for $a$, $b$, and $a+b$, finding nothing.

I suppose one could ask the same question for triples $a,b,c$ such that every integer up to $k$ is a factor of one of them, or quadruples, or....

The relation to question 98330 is as follows. With the notation used here, the sets $A=\lbrace a+1,b+1\rbrace$ and $B=\lbrace 1,a+b+1\rbrace$ are indistinguishable modulo $m$ for all $m$ up to $k$.

powersbetween $k$ and $n$. E.g., for $k=7$, neither $2L(7)=840$ nor $(11)(13)=143$ is a multiple of 9. – Gerry Myerson Jun 1 '12 at 2:01