A211316 -id:A211316 - cp's OEIS frontend

A211317 a(n) = A211316(2n+1).

0, 1, 2, 2, 3, 4, 4, 6, 6, 6, 7, 8, 10, 9, 10, 10, 12, 14, 12, 13, 14, 14, 18, 16, 16, 18, 18, 22, 19, 20, 20, 21, 26, 22, 24, 24, 24, 30, 28, 26, 27, 28, 34, 30, 30, 30, 31, 38, 32, 36, 34, 34, 42, 36, 36, 37, 38, 46, 39, 42, 44, 42, 50, 42, 43, 44, 44, 54, 46, 46, 48, 52, 58, 49, 50

Offset: 0

N. J. A. Sloane, Apr 24 2012

Considering only the odd-indexed terms of A211316 is motivated by the fact that A211316(2n) = n, so the idea is to skip these "uninteresting" terms. - M. F. Hasler, Jun 30 2025

Cf. A211316.

apply( A211317(n)=A211316(2*n+1), [0..88]) \\ M. F. Hasler, Jun 30 2025

Extended to a(0) = 0 and more terms from M. F. Hasler, Jun 30 2025

A289435 The arithmetic function v_3(n,3).

Original entry on oeis.org

1, 0, 2, 2, 3, 2, 4, 2, 5, 4, 6, 4, 7, 6, 8, 6, 9, 6, 10, 6, 11, 8, 12, 10, 13, 8, 14, 10, 15, 10, 16, 12, 17, 14, 18, 12, 19, 12, 20, 14, 21, 14, 22, 18, 23, 16, 24, 16, 25, 18, 26, 18, 27, 22, 28, 18, 29, 20, 30, 20, 31, 20, 32, 26, 33, 22, 34, 24, 35

Offset: 2

N. J. A. Sloane, Jul 06 2017

nonn

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

Cf. A211316 (equals v_1(n,3)).

a:= n-> n*max(seq((floor((d-1-igcd(d, 3))/3)+1)
        /d, d=numtheory[divisors](n))):
seq(a(n), n=2..100);  # Alois P. Heinz, Jul 07 2017

a[n_]:=n*Max[Table[(Floor[(d - 1 - GCD[d, 3])/3] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Indranil Ghosh, Jul 08 2017 *)

v(g,n,h)={my(t=0);fordiv(n,d,t=max(t,((d-1-gcd(d,g))\h + 1)*(n/d)));t}
a(n)=v(3,n,3); \\ Andrew Howroyd, Jul 07 2017

from sympy import divisors, floor, gcd
def a(n): return n*max((floor((d - 1 - gcd(d, 3))/3) + 1)/d for d in divisors(n))
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Jul 08 2017

a(41)-a(70) from Andrew Howroyd, Jul 07 2017

A007865 Number of sum-free subsets of {1, ..., n}.

Original entry on oeis.org

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

Offset: 0

Peter J. Cameron

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

			{} 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.

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

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..88, (terms up to a(70) from Per Hakan Lundow)
József Balogh, Hong Liu, Maryam Sharifzadeh, and Andrew Treglown, Sharp bound on the number of maximal sum-free subsets of integers, J. Eur. Math. Soc. 20 (2018), no. 8, pp. 1885-1911, also arXiv:1502.07605, Feb. 2015.
P. J. Cameron and P. Erdős, On the number of integers with various properties, in R. A. Mullin, ed., Number Theory: Proc. First Conf. of Canad. Number Theory Assoc. Conf., Banff, De Gruyter, Berlin, 1990, pp. 61-79.
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-Erdős conjecture, Discrete Math., 308 (2008), 4361-4369.
Eric Weisstein's World of Mathematics, Sum-Free Set

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.

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

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 *)

\\ 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

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

More terms from John W. Layman, Oct 21 2000
Extended through a(35) by Robert Israel, Nov 16 2005
a(36)-a(37) from Alec Mihailovs (alec(AT)mihailovs.com) (using Robert Israel's procedure), Nov 16 2005
a(38) from Eric W. Weisstein, Nov 17 2005
a(39)-a(42) from Eric W. Weisstein, Nov 28 2005, using Paul Abbott's Mathematica code

A290636 The arithmetic function v_3(n,4).

Original entry on oeis.org

1, 0, 2, 1, 3, 2, 4, 2, 5, 3, 6, 3, 7, 3, 8, 4, 9, 5, 10, 6, 11, 6, 12, 6, 13, 6, 14, 7, 15, 8, 16, 9, 17, 10, 18, 9, 19, 9, 20, 10, 21, 11, 22, 11, 23, 12, 24, 14, 25, 12, 26, 13, 27, 15, 28, 15, 29, 15, 30, 15, 31, 18, 32, 16, 33, 17, 34, 18, 35

Offset: 2

Robert Price, Aug 13 2017

nonn

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Robert Israel, Table of n, a(n) for n = 2..10000
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

Cf. A211316 (equals v_1(n,3)).
Cf. A289435 - A289441.
Cf. A024698.

f:= n -> max(map(t -> (floor((t - 1 - igcd(t, 3))/4) + 1)*n/t, numtheory:-divisors(n))):
map(f, [$2..100]); # Robert Israel, Aug 16 2017

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[3, n, 4], {n, 2, 70}]

For n >= 3, f(prime(n)) = A024698(n-1). - Robert Israel, Aug 16 2017

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A211317 a(n) = A211316(2n+1).

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A289435 The arithmetic function v_3(n,3).

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A007865 Number of sum-free subsets of {1, ..., n}.

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A290636 The arithmetic function v_3(n,4).

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Maple

Mathematica

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