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A007865 Number of sum-free subsets of {1, ..., n}. 46
1, 2, 3, 6, 9, 16, 24, 42, 61, 108, 151, 253, 369, 607, 847, 1400, 1954, 3139, 4398, 6976, 9583, 15456, 20982, 32816, 45417, 70109, 94499, 148234, 200768, 308213, 415543, 634270, 849877, 1311244, 1739022, 2630061, 3540355, 5344961, 7051789, 10747207, 14158720, 21295570, 28188520 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

More precisely, subsets of {1,...,n} containing no solutions of x+y=z.

There are two proofs that a(n) is 2^{n/2}(1+o(1)), as Paul Erdős and I conjectured.

In sumset notation, number of subsets A of {1,...,n} such that the intersection of A and 2A is empty. Using the Mathematica program, all such subsets can be printed. - T. D. Noe, Apr 20 2004

The Sapozhenko paper has many additional references.

The subsequence of primes begins: 2, 3, 61, 151, 607, 308213, 415543, no others through a(70). - Jonathan Vos Post, May 27 2010

If this sequence counts sum-free sets, then A326083 counts sum-closed sets, which is different from sum-full sets (A093971). - Gus Wiseman, Jul 08 2019

REFERENCES

P. J. Cameron and P. Erdős, On the number of integers with various properties, in R. A. Mullin, ed., Number Theory: Proc. First Con. of Canad. Number Theory Assoc. Conf., Banff, De Gruyter, Berlin, 1990, pp. 61-79.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 180-183.

LINKS

Per Hakan Lundow, Table of n, a(n) for n = 0..70

Steven R. Finch, Several Problems Concerning Sum-Free Sets [Broken link]

Steven R. Finch, Several Problems Concerning Sum-Free Sets [From the Wayback machine]

Ben Green and Imre Z. Ruzsa, Sum-free sets in abelian groups, arXiv:math/0307142 [math.CO], 2004.

A. A. Sapozhenko, The Cameron-Erdos conjecture, Discrete Math., 308 (2008), 4361-4369.

Eric Weisstein's World of Mathematics, Sum-Free Set

FORMULA

a(n) = A050291(n)-A088810(n) = A085489(n)-A088811(n) = A050291(n)+A085489(n)-A088813(n). - Reinhard Zumkeller, Oct 19 2003

EXAMPLE

{} has one sum-free subset, the empty set, so a(0)=1; {1} has two sum-free subsets, {} and {1}, so a(1)=2.

a(2) = 3: 0,1,2.

a(3) = 6: 0,1,2,3,13,23.

a(4) = 9: 0,1,2,3,4,13,14,23,34.

MAPLE

S3S:= {{}}: a[0]:= 1: for n from 1 to 35 do S3S:= S3S union map(t -> t union {n}, select(t -> (t intersect map(q -> n-q, t)={}), S3S)); a[n]:= nops(S3S) od: seq(a[n], n=0..35); # Code for computing (the number of) sum-free subsets of {1, ..., n} - Robert Israel

MATHEMATICA

SumFreeSet[ 0 ] = {{}}; SumFreeSet[ n_ ] := SumFreeSet[ n ] = Union[ SumFreeSet[ n - 1 ], Union[ #, {n} ] & /@ Select[ SumFreeSet[ n - 1 ], Intersection[ #, n - # ] == {} & ] ] As a check, enter Length /@ SumFreeSet /@ Range[ 0, 30 ] Alternatively, use NestList. n = 0; Length /@ NestList[ (++n; Union[ #, Union[ #, {n} ] & /@ Select[ #, Intersection[ #, n - # ] == {} & ] ]) &, {{}}, 30 ] (* from Paul Abbott, based on Robert Israel's Maple code *)

Timing[ n = 0; Last[ Reap[ Nest[ (++n; Sow[ Length[ # ] ]; Union[ #, Union[ #, {n} ]& /@ Select[ #, Intersection[ #, n - # ] == {} & ] ]) &, {{}}, 36 ] ] ] ] (* improved code from Paul Abbott, Nov 24 2005 *)

Table[Length[Select[Subsets[Range[n]], Intersection[#, Total/@Tuples[#, 2]]=={}&]], {n, 1, 10}] (* Gus Wiseman, Jul 08 2019 *)

CROSSREFS

See A085489 for another version.

Cf. A211316, A211317, A093970, A093971 (number of sum-full subsets of 1..n).

Cf. A050291, A103580, A151897, A308546, A326020, A326080, A326083.

Sequence in context: A147364 A147227 A147063 * A052812 A213331 A218153

Adjacent sequences:  A007862 A007863 A007864 * A007866 A007867 A007868

KEYWORD

nonn,nice

AUTHOR

Peter J. Cameron

EXTENSIONS

More terms from John W. Layman, Oct 21 2000

Extended through 2630061 by Robert Israel, Nov 16 2005; two further terms from Alec Mihailovs (alec(AT)mihailovs.com) (using Robert Israel's procedure), Nov 16 2005

7051789 from Eric W. Weisstein, Nov 17 2005

10747207, 14158720, 21295570, 28188520 from Eric W. Weisstein, Nov 28 2005, using Paul Abbott's Mathematica code

STATUS

approved

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Last modified September 21 14:54 EDT 2020. Contains 337272 sequences. (Running on oeis4.)