A356368 - OEIS

%I #16 Oct 19 2022 05:53:33

%S 88,98,99,110,163,177,178

%N Sparse ruler lengths with unique non-Wichmann solutions.

%C These are the only proven unique non-Wichmann sparse rulers:{0,1,2,8,15,16,26,36,46,56,66,76,79,83,85,88},

%C   {0,1,2,8,15,16,26,36,46,56,66,76,86,89,93,95,98},

%C   {0,1,2,5,10,15,20,25,36,52,58,69,85,91,97,98,99},

%C   {0,1,2,5,10,15,20,25,36,52,63,69,80,96,102,108,109,110},

%C   {0,1,3,10,19,28,32,37,47,62,77,92,107,122,137,143,149,155,157,160,161,163},

%C   {0,1,3,6,7,13,19,25,40,55,70,85,100,115,130,141,145,150,159,168,173,176,177},

%C   {0,1,3,10,19,28,32,37,47,62,77,92,107,122,137,152,158,164,170,172,175,176,178}.

%C Values with a single known sparse ruler include 334, 335, 385, 408, 426, 427, 449, 450, 473, 475, 560, 583, 608, 610. Cf. A326499 for representations.

%H J. Leech, <a href="https://doi.org/10.1112/jlms/s1-31.2.160">On the representation of 1, 2, ..., n by differences</a>, J. Lond. Math. Soc. 31 (1956), 160-169.

%H Peter Luschny, <a href="http://www.luschny.de/math/rulers/rulercnt.html">Perfect and Optimal Rulers</a>

%H Peter Luschny, <a href="http://www.luschny.de/math/rulers/optimalconjecture.html">Are optimal rulers of Wichmann type?</a>

%H Peter Luschny, <a href="http://www.luschny.de/math/rulers/prulers.html">Perfect Rulers.</a>

%H Peter Luschny, <a href="http://www.luschny.de/math/rulers/optimalconjecture.html">Wichmann Rulers.</a>

%H Ed Pegg Jr., <a href="http://demonstrations.wolfram.com/SparseRulers/">Sparse Rulers</a> (Wolfram Demonstrations Project)

%H Ed Pegg Jr., <a href="http://demonstrations.wolfram.com/WichmannLikeRulers/">Wichmann-like Rulers</a> (Wolfram Demonstrations Project)

%H Ed Pegg Jr, <a href="/A326499/a326499_1.txt">Table of n, a(n) for n=1..10501 in batches of A289761.</a> Transpose for Dark Mills pattern.

%H Ed Pegg Jr, <a href="/A326499/a326499_2.txt">Sparse rulers and excess values for lengths n=1..10501</a>.

%H Ed Pegg Jr, <a href="/A326499/a326499.jpg">Picture of a(n) for n = 1..10501 in batches of A289761</a>. This is the Dark Mills pattern.

%H L. Rédei, A. Rényi, <a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=sm&amp;paperid=5985&amp;option_lang=eng">On the representation of the numbers 1, 2, ..., N by means of differences</a>, Matematicheskii Sbornik, Vol. 24(66) Num. 3 (1949), 385-389 (in Russian).

%H B. Wichmann, <a href="https://doi.org/10.1112/jlms/s1-38.1.465">A note on restricted difference bases</a>, J. Lond. Math. Soc. 38 (1963), 465-466.

%Y Cf. A046693, A289761, A308766, A309407, A326499.

%K nonn,hard,more

%O 1,1

%A _Ed Pegg Jr_, Oct 16 2022