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We study numerical semigroups $S$ with the property that if $m$ is the multiplicity of $S$ and $w\left(i\right)$ is the least element of $S$ congruent with $i$ modulo $m$, then $0<w\left(1\right)<\cdots <w(m-1)$. The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups.