  We study almost symmetric semigroups generated by odd integers. If the
embedding dimension is four, we characterize when a symmetric semigroup that is
not complete intersection or a pseudo-symmetric semigroup is generated by odd
integers. Moreover, we give a way to construct all the almost symmetric
semigroups with embedding dimension four and type three generated by odd
elements. In this case we also prove that all the pseudo-Frobenius numbers are
multiple of one of them and this gives many consequences on the semigroup and
its defining ideal.
