A104309 - OEIS

%I #11 Feb 23 2021 12:27:54

%S 1,3,5,7,10,12,14,16,18,20,24,24,27,30,31,33,37,37,39,44,44,45,51,51,

%T 51,54,59,59,60,62,69,69,69,70,80,80,80,81,83,91,91,91,91,93

%N Minimum length of a perfect ruler that contains a segment not shorter than n.

%C For nomenclature related to perfect and optimal rulers see Peter Luschny's "Perfect Rulers" web pages.

%H F. Schwartau, Y. Schröder, L. Wolf and J. Schoebel, <a href="/A104309/b104309.txt">Table of n, a(n) for n = 1..92</a>

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

%H Hugo Pfoertner, <a href="http://www.randomwalk.de/scimath/diffset/consdifs.txt">Largest and smallest maximum differences of consecutive marks of perfect rulers.</a>

%H F. Schwartau, Y. Schröder, L. Wolf and J. Schoebel, <a href="https://dx.doi.org/10.21227/cd4b-nb07">MRLA search results and source code</a>, Nov 6 2020.

%H F. Schwartau, Y. Schröder, L. Wolf and J. Schoebel, <a href="https://doi.org/10.1109/OJAP.2020.3043541">Large Minimum Redundancy Linear Arrays: Systematic Search of Perfect and Optimal Rulers Exploiting Parallel Processing</a>, IEEE Open Journal of Antennas and Propagation, 2 (2021), 79-85.

%H <a href="/index/Per#perul">Index entries for sequences related to perfect rulers.</a>

%e The list of shortest perfect rulers containing a segment>=n starts:

%e n.a(n)..rulers..(marks enclosing longest segment)

%e 1..1....[0,1]........(0,1)

%e 2..3....[0,1,3]......(1,3)

%e 3..5....[0,1,2,5]....(2,5)

%e 4..7....[0,1,2,3,7]..(3,7)

%e 5.10....[0,1,2,4,9,10]..(4,9)

%e ........[0,1,3,4,9,10]..(4,9)

%e ........[0,1,6,7,8,10]..(1,6)

%e 6.12....[0,1,3,5,11,12]..(5,11)

%e ........[0,1,7,8,10,12]..(1,7)

%e 7.14....[0,1,2,4,6,13,14]...(6,13)

%e ........[0,1,3,4,6,13,14]...(6,13)

%e ........[0,1,3,5,6,13,14]...(6,13)

%e ........[0,1,8,9,10,12,14]..(1,8)

%e ........[0,1,8,9,11,12,14]..(1,8)

%e 8.16....[0,1,3,5,7,15,16]....(7,15)

%e ........[0,1,9,10,12,14,16]..(1,9)

%Y Cf. A104305 largest possible segment in a perfect ruler of length n, A104310 maximum length of perfect rulers made from segments not exceeding n, A103294 definitions related to complete rulers.

%K hard,nonn

%O 1,2

%A _Hugo Pfoertner_, Mar 01 2005