A140794 One of the four smallest counterexamples to the conjecture that the cardinality of the sumset is less than or equal to the cardinality of the difference set of every finite set of integers.

0, 2, 3, 7, 10, 11, 12, 14

Offset: 1

Author

Jonathan Vos Post, Jul 15 2008

Keywords

• fini
• full
• nonn

Comments

This sequence is the reflection of A102282: a(n) = 14 - A102282(9-n).

Keywords: sum-dominant sets, MSTD sets.

A set with more sums than differences is called an MSTD set. Hegarty has constructed many such examples.

Comment from N. J. A. Sloane, Mar 10 2013: Out of the 2^n subsets S of [0..n-1], let

AG(n) = number of S with |S+S|>|S-S|,

AE(n) = number of S with |S+S|=|S-S|,

AL(n) = number of S with |S+S|<|S-S|.

A140794 says AG(n) = 0 for n <= 14. These three sequences are respectively A222807, A118544, A222808.

Examples

			Let A = {0, 2, 3, 7, 10, 11, 12, 14}. Then the cardinality of the sumset, |A + A| = 26, while the cardinality of the difference set, |A - A| = 25.
		

Links

• P. V. Hegarty, Some explicit constructions of sets with more sums than differences, Acta Arith., 130 (2007), 61-77.
• Greg Martin and Kevin O'Bryant, Many sets have more sums than differences, arXiv:math/0608131 [math.NT], 2006.
• Melvyn B. Nathanson, Problems in Additive Number Theory, III: Thematic Seminars at the Centre de Recerca Matematica, arXiv:0807.2073 [math.NT], 2008.

Crossrefs

Cf. A222807, A118544, A222808.

Extensions

Corrected by James Wilcox, Jul 24 2013