  A word $u=u_1\dots u_n$ is a scattered factor of a word $w$ if $u$ can be
obtained from $w$ by deleting some of its letters: there exist the (potentially
empty) words $v_0,v_1,..,v_n$ such that $w = v_0u_1v_1...u_nv_n$. The set of
all scattered factors up to length $k$ of a word is called its full
$k$-spectrum. Firstly, we show an algorithm deciding whether the $k$-spectra
for given $k$ of two words are equal or not, running in optimal time. Secondly,
we consider a notion of scattered-factors universality: the word $w$, with
$\letters(w)=\Sigma$, is called $k$-universal if its $k$-spectrum includes all
words of length $k$ over the alphabet $\Sigma$; we extend this notion to
$k$-circular universality. After a series of preliminary combinatorial results,
we present an algorithm computing, for a given $k'$-universal word $w$ the
minimal $i$ such that $w^i$ is $k$-universal for some $k>k'$. Several other
connected problems~are~also~considered.
