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# Multiplicative sequences

${\displaystyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(mn)\,=\,a(m)\,a(n),\,}$
where ${\displaystyle \scriptstyle m\,\perp \,n\,}$ means ${\displaystyle \scriptstyle m\,}$ coprime (or orthogonal) to ${\displaystyle \scriptstyle n.\,}$
Since ${\displaystyle \scriptstyle a(n)\,=\,a(1n)\,=\,a(1)\,a(n)\,}$, we then have ${\displaystyle \scriptstyle a(1)\,=\,1.\,}$