

A263996


Smallest possible cardinality of the union of the set of pairwise sums and the set of pairwise products from a set of n positive integers.


2



1, 4, 7, 11, 15, 20, 26, 30, 36, 44, 49, 57, 64, 71, 80, 86, 96, 104, 112, 121, 131, 141, 150, 160, 169, 179, 190, 200, 212, 222, 235, 248, 260, 272, 283, 296, 307, 320, 335, 348, 360, 371
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OFFSET

1,2


COMMENTS

The November 2015  February 2016 round of Al Zimmermann's programming contests asked for optimal sets producing a(40), a(80), a(120), ..., a(1000).


REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., SpringerVerlag New York, 2004. Problem F18.


LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..205
P. Erdős and E. Szemeredi, On sums and products of integers, Studies in Pure Mathematics, Birkhäuser, Basel, 1983, pp. 213218. DOI:10.1007/9783034854382_19
Al Zimmermann's Programming Contests, Sums and Products, Nov 2015  Feb 2016.


EXAMPLE

a(1) = 1 because for the set {2} the union of {2+2} and {2*2} = {4}.
a(7) = 26: The set {1,2,3,4,6,8,12} has the set of pairwise sums {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,24} and the set of pairwise products {1,2,3,4,6,8,9,12,16,18,24,32,36,48,64,72,96,144}. The cardinality of the union of the two sets, {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,24,32,36,48,64,72,96,144}, is 26. This is the first nontrivial case with a(n) < A263995(n), which uses the set {1..n}.


CROSSREFS

Cf. A263995.
Sequence in context: A130252 A130254 A278114 * A172472 A134918 A310741
Adjacent sequences: A263993 A263994 A263995 * A263997 A263998 A263999


KEYWORD

nonn


AUTHOR

Hugo Pfoertner, Nov 15 2015


STATUS

approved



