A385931 a(n) is the least integer such that there is an n-element set of integers between 0 and a(n) such that the sums of any three distinct elements are all distinct.

0, 1, 2, 3, 7, 13, 22, 39, 69, 113

Offset

1,3

Comments

Such sets are related to Sidon sets and could be called weak B_3 sets.

Links

Table of n, a(n) for n=1..10.

Example

Subsets for n=3..8:
  {0, 1, 2},
  {0, 1, 2, 3},
  {0, 1, 2, 4, 7},
  {0, 1, 2, 4, 7, 13},
  {0, 1, 2, 10, 16, 19, 22},
  {0, 1, 2, 10, 19, 33, 36, 39}.
For some n there are multiple subsets, as for example at n=6 also {0,1,2,7,10,13}.

Prog

(PARI)
a(n)={
  my(recurse(k, j, b, b2, b3)=
    if(!k, 1, (j>k && self()(k, j-1, b, b2, b3))
    || (j>0 && bitand(b2, b<<j)==0 && bitand(b3, b2<<j)==0 && self()(k-1, j-1, b+(1<<j), b2+(b<<j), b3+(b2<<j))))
  );
  if(n<=4, n-1, for(j=2, oo, if(recurse(n-2, j-1, (1<<j)+1, 1<<j, 0), return(j))))
} \\ Andrew Howroyd, Sep 13 2025

Crossrefs

Cf. A345731.

Sequence in context: A048216 A003509 A394768 * A238432 A238423 A133370

Adjacent sequences: A385928 A385929 A385930 * A385932 A385933 A385934

Keyword

nonn,hard,more

Author

Wei-Hwa Huang, Jul 12 2025

Extensions

a(9)-a(10) from Andrew Howroyd, Sep 13 2025

Status

approved