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).

1, 3, 5, 9, 13, 17, 21, 27, 33, 41, 47, 55, 65, 73, 81, 93, 105, 117, 129, 141, 153, 165, 181, 197, 213

Offset: 1

Author

Warren D. Smith, Oct 02 2006

Keywords

• hard
• more
• nonn

Comments

Notation: N[q] = the set of q+1 elements inside {0,1,...,N-1}

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

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.

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

Examples

			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.
N[q]: set
------------------------------
3[2]: 0,1,
4[3]: 0,1,2,
5[3]: 0,1,2,
6[3]: 0,2,3,
7[4]: 0,1,2,3,
8[4]: 0,1,3,4,
9[4]: 0,1,3,4,
10[5]: 0,1,2,4,5,
11[5]: 0,1,2,4,5,
12[5]: 0,1,3,5,6,
13[5]: 0,1,3,5,6,
14[6]: 0,1,2,4,6,7,
15[6]: 0,1,2,4,6,7,
16[6]: 0,1,3,5,7,8,
17[6]: 0,1,3,5,7,8,
18[6]: 0,2,3,7,8,10,
19[7]: 0,1,2,4,6,8,9,
20[7]: 0,1,3,5,7,9,10,
21[7]: 0,1,3,5,7,9,10,
22[7]: 0,2,3,7,8,10,11,
23[8]: 0,1,2,4,6,8,10,11,
24[8]: 0,1,3,5,7,9,11,12,
25[8]: 0,1,3,5,7,9,11,12,
26[8]: 0,2,3,7,8,10,12,13,
27[8]: 0,1,3,4,9,10,12,13,
28[8]: 0,2,3,7,8,12,13,15,
29[9]: 0,1,3,5,7,9,11,13,14,
30[9]: 0,2,3,7,8,10,12,14,15,
31[9]: 0,1,3,4,9,10,12,14,15,
32[9]: 0,2,3,7,8,12,13,15,16,
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.
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
		

Links

• S. Gunturk and M. B. Nathanson, A new upper bound for finite additive bases, Acta Arithmetica, Vol. 124, No. 3 (2006) 235-255.
• Jürgen Herzog, Shinya Kumashiro, and Dumitru I. Stamate, The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one, arXiv:2106.09404 [math.AC], 2021. See p. 9.
• Jürgen Herzog, Shinya Kumashiro, and Dumitru I. Stamate, The far-flung Gorenstein property for numerical semigroups, Extended abstract for IMNS 2024. See p. 3.
• Kagawa, VBA program
• H. Rohrbach, Ein Beitrag zur additiven Zahlentheorie, Math. Z. 42 (1937) 1-30. DOI
• W. D. Smith, More information

Crossrefs

Cf. A001212, A008932.

Formula

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.

lim inf a(n) / n^2 > 0.2857 lim sup a(n) / n^2 < 0.4789 - Charles R Greathouse IV, Aug 11 2007

Extensions

More terms (from Smith's web site) from R. J. Mathar, Oct 08 2006

Entry revised by N. J. A. Sloane, Aug 06 2017

a(13)-a(25) from Herzog et al. added by Stefano Spezia, Jul 05 2024