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A123509 Rohrbach's problem: a(n) is the largest integer such that there exists a set of n integers that is a basis of order 2 for (0, 1, ..., a(n)-1).

%I A123509 #58 Aug 11 2025 04:08:15
%S A123509 1,3,5,9,13,17,21,27,33,41,47,55,65,73,81,93,105,117,129,141,153,165,
%T A123509 181,197,213
%N A123509 Rohrbach's problem: a(n) is the largest integer such that there exists a set of n integers that is a basis of order 2 for (0, 1, ..., a(n)-1).
%C A123509 Notation: N[q] = the set of q+1 elements inside {0,1,...,N-1}
%C A123509 Length of the longest sequence of consecutive integers that can be obtained from a set of n distinct integers by summing any two integers in the set or doubling any one. - _Jon E. Schoenfield_, Jul 16 2017
%C A123509 According to Zhining Yang, Jul 08 2017, a(13) to a(20) are 65, 70, 79, 90, 101, 112, 123, 134, but there is some doubt about these terms, and they should be confirmed before they are accepted. They do not agree with the conjecture, so perhaps the VBA program is not correct.
%C A123509 The definition of Rohrbach's Problem in the paper of S. Gunturk and M. B. Nathanson in the links is different from the one here. In the paper, the set should contain n nonnegative integers instead of integers. The result should be equal to A001212(n-1)+1 according to the definition in the paper since adding one 0 before any set for A001212(n-1) provides a set of the problem. The data provided by Zhining Yang is obviously wrong since a(n) >= A001212(n-1)+1. And A302648 provides another lower bound of this array since a(n) >= 2*A302648(n)+1. - _Zhao Hui Du_, Apr 13 2018
%H A123509 S. Gunturk and M. B. Nathanson, <a href="http://dx.doi.org/10.4064/aa124-3-3">A new upper bound for finite additive bases</a>, Acta Arithmetica, Vol. 124, No. 3 (2006) 235-255.
%H A123509 Jürgen Herzog, Shinya Kumashiro, and Dumitru I. Stamate, <a href="https://arxiv.org/abs/2106.09404">The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one</a>, arXiv:2106.09404 [math.AC], 2021. See p. 9.
%H A123509 Jürgen Herzog, Shinya Kumashiro, and Dumitru I. Stamate, <a href="https://imns2024.uca.es/wp-content/uploads/2024/06/Dumitru-Stamate.pdf">The far-flung Gorenstein property for numerical semigroups</a>, Extended abstract for IMNS 2024. See p. 3.
%H A123509 Kagawa, <a href="http://club.excelhome.net/forum.php?mod=redirect&amp;goto=findpost&amp;ptid=1355454&amp;pid=9161208">VBA program</a>
%H A123509 H. Rohrbach, <a href="https://eudml.org/doc/168701">Ein Beitrag zur additiven Zahlentheorie</a>, Math. Z. 42 (1937) 1-30. <a href="https://doi.org/10.1007/BF01160061">DOI</a>
%H A123509 W. D. Smith, <a href="http://rangevoting.org/Sum2cov.html">More information</a>
%F A123509 a(n) = A001212(n-1)+1 (conjecture). - _R. J. Mathar_, Oct 08 2006. Comment from _Martin Fuller_, Mar 18 2009: I agree with this conjecture.
%F A123509 lim inf a(n) / n^2 > 0.2857 lim sup a(n) / n^2 < 0.4789 - _Charles R Greathouse IV_, Aug 11 2007
%e A123509 Example: 8[3]: 0,1,3,4 means {0,1,2,...,8} is covered thus: 0=0+0, 1=0+1, 2=1+1, 3=0+3, 4=0+4=1+3, 5=1+4, 6=3+3, 7=3+4, 8=4+4.
%e A123509 N[q]: set
%e A123509 ------------------------------
%e A123509 3[2]: 0,1,
%e A123509 4[3]: 0,1,2,
%e A123509 5[3]: 0,1,2,
%e A123509 6[3]: 0,2,3,
%e A123509 7[4]: 0,1,2,3,
%e A123509 8[4]: 0,1,3,4,
%e A123509 9[4]: 0,1,3,4,
%e A123509 10[5]: 0,1,2,4,5,
%e A123509 11[5]: 0,1,2,4,5,
%e A123509 12[5]: 0,1,3,5,6,
%e A123509 13[5]: 0,1,3,5,6,
%e A123509 14[6]: 0,1,2,4,6,7,
%e A123509 15[6]: 0,1,2,4,6,7,
%e A123509 16[6]: 0,1,3,5,7,8,
%e A123509 17[6]: 0,1,3,5,7,8,
%e A123509 18[6]: 0,2,3,7,8,10,
%e A123509 19[7]: 0,1,2,4,6,8,9,
%e A123509 20[7]: 0,1,3,5,7,9,10,
%e A123509 21[7]: 0,1,3,5,7,9,10,
%e A123509 22[7]: 0,2,3,7,8,10,11,
%e A123509 23[8]: 0,1,2,4,6,8,10,11,
%e A123509 24[8]: 0,1,3,5,7,9,11,12,
%e A123509 25[8]: 0,1,3,5,7,9,11,12,
%e A123509 26[8]: 0,2,3,7,8,10,12,13,
%e A123509 27[8]: 0,1,3,4,9,10,12,13,
%e A123509 28[8]: 0,2,3,7,8,12,13,15,
%e A123509 29[9]: 0,1,3,5,7,9,11,13,14,
%e A123509 30[9]: 0,2,3,7,8,10,12,14,15,
%e A123509 31[9]: 0,1,3,4,9,10,12,14,15,
%e A123509 32[9]: 0,2,3,7,8,12,13,15,16,
%e A123509 a(5)=13 because we can obtain at most a total of 13 consecutive integers from a set of 5 integers by summing any two integers in the set or doubling any one; from the 5-integer set {1,2,4,6,7}, we can obtain all 13 integers in the interval [2..14] as follows: 2=1+1, 3=1+2, 4=2+2, 5=1+4, 6=2+4, 7=1+6, 8=2+6, 9=2+7, 10=4+6, 11=4+7, 12=6+6, 13=6+7, 14=7+7.
%e A123509 a(16)=90 because we can obtain at most a total of 90 consecutive integers from a set of 16 integers by summing any two integers in the set or doubling any one: from the 16-integer set {1,2,4,5,8,9,10,17,18,22,25,36,47,58,69,80}, we can obtain all 90 integers in the interval [2..91]. - _Jon E. Schoenfield_, Jul 16 2017
%Y A123509 Cf. A001212, A008932.
%K A123509 hard,more,nonn
%O A123509 1,2
%A A123509 _Warren D. Smith_, Oct 02 2006
%E A123509 More terms (from Smith's web site) from _R. J. Mathar_, Oct 08 2006
%E A123509 Entry revised by _N. J. A. Sloane_, Aug 06 2017
%E A123509 a(13)-a(25) from Herzog et al. added by _Stefano Spezia_, Jul 05 2024