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A007865
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Number of sum-free subsets of {1, ..., n}.
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104
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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
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OFFSET
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0,2
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COMMENTS
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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.
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
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 180-183.
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LINKS
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FORMULA
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EXAMPLE
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{} 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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) \\ good only for n <= 25:
sumfree(v) = {for(i=1, #v, for (j=1, i, if (setsearch(v, v[i]+v[j]), return (0)); ); ); return (1); }
a(n) = {my(nb = 0); forsubset(n, s, if (sumfree(Set(s)), nb++); ); nb; } \\ Michel Marcus, Nov 08 2020
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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a(36)-a(37) from Alec Mihailovs (alec(AT)mihailovs.com) (using Robert Israel's procedure), Nov 16 2005
a(39)-a(42) from Eric W. Weisstein, Nov 28 2005, using Paul Abbott's Mathematica code
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STATUS
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approved
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