For some number of marks, what is the longest length with that many marks that is a sparse ruler? In [1], Wichmann introduced what is now called the Wichmann ruler (rule 1 in this Demonstration) with differences (or mark spacing) given by

. With , , that gives as a split form.

The rule means . With , , the numbers produced are as a differences form.

For all numbers of the form , one of these original Wichmann rulers exists, with marks and length . There are also many Wichmann-like variants that cover other lengths. The 2340 Wichmann variants in this Demonstration all produce infinitely many sparse rulers, including more than a hundred thousand sparse rulers under length 4444.

For a strictly increasing integer subset with elements, define the sparseness as one of two things: If any differences 1 to are missing, is the list of missing differences. With nothing missing, . Up to length at least 1750, a sparse ruler always exists with sparseness .

For the first 886 rules, no differences are ever missing. Are there more Wichmann-like rulers of this type?