Multiplicity sequences

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Multiplicity sequences are sequences with the property

\forall \,m,\,n\,\in \,\mathbb {N} ,\,{\rm {lcm}}(a(m),\,a(n))=a({\rm {lcm}}(m,\,n)),\,

where $\scriptstyle {\rm {lcm}}(m,\,n)\,$ is the least common multiple (lcm) of $\scriptstyle m\,$ and $\scriptstyle n.\,$

Additive sequences and Category:Additive sequences ( $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(mn)\,=\,a(m)+a(n)\,$ $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(mn)\,=\,a(m)+a(n)\,$ )
- Completely additive sequences and Category:Completely additive sequences ( $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,a(mn)\,=\,a(m)+a(n)\,$ )

Multiplicative sequences and Category:Multiplicative sequences ( $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(mn)\,=\,a(m)\,a(n)\,$ $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(mn)\,=\,a(m)\,a(n)\,$ )
- Completely multiplicative sequences and Category:Completely multiplicative sequences ( $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,a(mn)\,=\,a(m)\,a(n)\,$ )

Coprimality sequences and Category:Coprimality sequences ( $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(m)\,\perp \,a(n)\,$ $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(m)\,\perp \,a(n)\,$ )
- Coprime sequences and Category:Coprime sequences ( $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\neq \,n\,\implies \,a(m)\,\perp \,a(n)\,$ )

Divisibility sequences and Category:Divisibility sequences ( $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,|\,n\,\implies \,a(m)\,|\,a(n)\,$ )
Strong divisibility sequences and Category:Strong divisibility sequences ( $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,\gcd(a(m),\,a(n))\,=\,a(\gcd(m,\,n))\,$ )
Multiplicity sequences and Category:Multiplicity sequences ( $\scriptstyle \forall \,m,\,n\,\in \,\mathbb {N} ,\,{\rm {lcm}}(a(m),\,a(n))\,=\,a({\rm {lcm}}(m,\,n))\,$ )

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