# Sidon sets and their generalizations, Sidon $$g$$-sets¶

AUTHORS:

• Martin Raum (07-25-2011)

sage.combinat.sidon_sets.sidon_sets(N, g=1)

Return the set of all Sidon-$$g$$ sets that have elements less than or equal to $$N$$.

A Sidon-$$g$$ set is a set of positive integers $$A \subset [1, N]$$ such that any integer $$M$$ can be obtain at most $$g$$ times as sums of unordered pairs of elements of $$A$$ (the two elements are not necessary distinct):

$\#\{ (a_i, a_j) | a_i, a_j \in A, a_i + a_j = M,a_i \leq a_j \} \leq g$

INPUT:

• $$N$$ – A positive integer.

• $$g$$ – A positive integer (default: $$1$$).

OUTPUT:

• A Sage set with categories whose element are also set of integers.

EXAMPLES:

sage: S = sidon_sets(3, 2)
sage: sorted(S, key=str)
[{1, 2, 3}, {1, 2}, {1, 3}, {1}, {2, 3}, {2}, {3}, {}]
sage: S.cardinality()
8
sage: S.category()
Category of finite sets
sage: sid = S.an_element()
sage: sid
{2}
sage: sid.category()
Category of finite sets

sage.combinat.sidon_sets.sidon_sets_rec(N, g=1)

Return the set of all Sidon-$$g$$ sets that have elements less than or equal to $$N$$ without checking the arguments. This internal function should not be call directly by user.