Wichmann-like Rulers

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A sparse ruler is a rod of integer length with a minimal number of marks so that all distances 1 to can be measured. With 1 to 9, its five marks are at with differences between marks . The differences 1 to 9 are , , , , , , , , .

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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?

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Contributed by: Ed Pegg Jr (August 2019)
Open content licensed under CC BY-NC-SA


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References

[1] B. Wichmann, "A Note on Restricted Difference Bases," Journal of the London Mathematical Society, s1-38(1), 1963 pp. 465–466. doi:10.1112/jlms/s1-38.1.465.

[2] A. D. Robison. "Parallel Computation of Sparse Rulers." (Aug 7, 2019) software.intel.com/en-us/articles/parallel-computation-of-sparse-rulers.

[3] P. Luschny. "Perfect and Optimal Rulers." (Aug 7, 2019) oeis.org/wiki/User:Peter_Luschny/PerfectRulers.


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