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

1,2

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.

Links

Table of n, a(n) for n=1..8.

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.

Example

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.

Crossrefs

Cf. A222807, A118544, A222808.

Sequence in context: A059180 A051637 A051471 * A047531 A102808 A215937

Adjacent sequences: A140791 A140792 A140793 * A140795 A140796 A140797

Keyword

fini,full,nonn

Author

Jonathan Vos Post, Jul 15 2008

Extensions

Corrected by James Wilcox, Jul 24 2013

Status

approved