A211316 Maximal size of sum-free set in additive group of integers mod n.

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

Offset

2,3

References

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, Manuscript, May 2017. See Table in Section 1.6.1.

A. P. Street, Counting non-isomorphic sum-free sets, in Proc. First Australian Conf. Combinatorial Math., Univ. Newcastle, 1972, pp. 141-143.

Links

Reinhard Zumkeller, 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.

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

Formula

If n is divisible by a prime == 2 mod 3 then a(n) = n(p+1)/(3p) where p is the smallest such prime divisor; otherwise if n is divisible by 3 then a(n) = n/3; otherwise all prime divisors of n are == 1 mod 3 and a(n) = (n-1)/3.

In particular, a(2n) = n (cf. A211317).

Mathematica

a[n_] := Module[{f = FactorInteger[n][[All, 1]]}, For[i = 1, i <= Length[f], i++, If[Mod[f[[i]], 3]==2, Return[n*(f[[i]] + 1)/3/f[[i]]]]]; If[Mod[n, 3] == 1, n-1, n]/3]
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Aug 02 2018, from PARI *)

Prog

(Haskell)
a211316 n | not $ null ps = n * (head ps + 1) `div` (3 * head ps)
          | m == 0        = n'
          | otherwise     = (n - 1) `div` 3
          where ps = [p | p <- a027748_row n, mod p 3 == 2]
                (n', m) = divMod n 3
-- Reinhard Zumkeller, Apr 25 2012
(PARI) a(n)=my(f=factor(n)[, 1]); for(i=1, #f, if(f[i]%3==2, return(n*(f[i]+1)/3/f[i]))); if(n%3, n-1, n)/3 \\ Charles R Greathouse IV, Sep 02 2015

Crossrefs

Bisection: A211317. Cf. A007865, A027748, A003627.

Sequence in context: A178804 A322355 A242112 * A280226 A307995 A061889

Adjacent sequences: A211313 A211314 A211315 * A211317 A211318 A211319

Keyword

nonn

Author

N. J. A. Sloane, Apr 24 2012

Status

approved