A102282 -id:A102282 - cp's OEIS frontend

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

Jonathan Vos Post, Jul 15 2008

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.

			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.

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.

Cf. A222807, A118544, A222808.

Corrected by James Wilcox, Jul 24 2013

A327819 Elements of the unique smallest MSTD set of primes.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 23, 43, 47, 53, 59, 61, 67, 71, 73

Offset: 1

Charles R Greathouse IV, Sep 26 2019

This set is smallest in terms of minimizing the maximum element.

Chu, McNew, Miller, Xu, & Zhang show that, as a consequence of the Green-Tao theorem, there are infinitely many MSTD sets of primes, and give an example. Chu finds this set and proves minimality, answering a question of the former authors.

Hung Chu, Nathan McNew, Steven J. Miller, Victor Xu, and Sean Zhang, When sets can and cannot have MSTD subsets (2016) arXiv:1608.03256 [math.NT]
Hung Viet Chum, Sets of cardinality 6 are not sum-dominant, arXiv:1906.00470 [math.NT], 2019.

Cf. A102282, A222807, A000040.

cp's OEIS Frontend

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.

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