This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A360029 #25 Feb 19 2023 09:17:12
%S A360029 1,3,6,10,15,18,25,33,42,52,63,71,84,98,107,123,140,152,171,185,198,
%T A360029 220,243,256,281,307,334,354,383,403,434,466,489,523,552,581,618,656,
%U A360029 695,728
%N A360029 Consider a ruler composed of n segments with lengths 1, 1/2, 1/3, ..., 1/n with total length A001008(n)/A002805(n). a(n) is the minimum number of distinct distances of all pairs of marks that can be achieved by permuting the positions of the segments.
%C A360029 Without permutation of the arrangement of the segments, the number of distinct distances between any pair of marks is n*(n+1)/2.
%H A360029 Hugo Pfoertner, <a href="/A360029/a360029.txt">Examples of rulers with the minimum number of measurable distances up to n=38</a>, Feb 01 2023.
%e A360029 a(6) = 18: permuted segment lengths 1, 1/4, 1/2, 1/3, 1/6, 1/5 -> marks at 0, 1, 5/4, 7/4, 25/12, 9/4, 49/20 -> 18 distinct distances 1/6, 1/5, 1/4, 1/3, 11/30, 1/2, 7/10, 3/4, 5/6, 1, 13/12, 6/5, 5/4, 29/20, 7/4, 25/12, 9/4, 49/20, whereas the non-permuted ruler with marks at 0, 1, 3/2, 11/6, 25/12, 137/60, 49/20 gives 21 distinct distances 1/6, 1/5, 1/4, 1/3, 11/30, 9/20, 1/2, 7/12, 37/60, 47/60, 5/6, 19/20, 1, 13/12, 77/60, 29/20, 3/2, 11/6, 25/12, 137/60, 49/20.
%o A360029 (PARI) a360029(n) = {if (n<=1, 1, my (mi=oo); w = vectorsmall(n-1, i, i+1);
%o A360029 forperm (w, p, my(v=vector(n,i,1/i), L=List(v)); for (m=1, n, v[m] = 1 + sum (k=1, m-1, 1/p[k]); listput(L, v[m])); for (i=1, n-1, for (j=i+1, n, listput (L, v[j]-v[i]))); mi = min(mi, #Set(L))); mi)};
%Y A360029 Cf. A000217, A001008, A002805, A003022.
%K A360029 nonn,hard,more
%O A360029 1,2
%A A360029 _Hugo Pfoertner_, Jan 22 2023
%E A360029 a(39)-a(40) from _Hugo Pfoertner_, Feb 19 2023