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# Multiplicative sequences

Multiplicative sequences are sequences with the property

$\forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(mn)\,=\,a(m)\,a(n),\,$ where $m\,\perp \,n\,$ means $m\,$ coprime (or orthogonal) to $n.\,$ Since $a(n)\,=\,a(1n)\,=\,a(1)\,a(n)\,$ , we then have $a(1)\,=\,1.\,$ • Additive sequences and Category:Additive sequences ($\forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(mn)\,=\,a(m)+a(n)\,$ )
• Coprimality sequences and Category:Coprimality sequences ($\forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(m)\,\perp \,a(n)\,$ )
• Coprime sequences and Category:Coprime sequences ($\forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\neq \,n\,\implies \,a(m)\,\perp \,a(n)\,$ )
• Divisibility sequences and Category:Divisibility sequences ($\forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,|\,n\,\implies \,a(m)\,|\,a(n)\,$ )
• Strong divisibility sequences and Category:Strong divisibility sequences ($\forall \,m,\,n\,\in \,\mathbb {N} ,\,\gcd(a(m),\,a(n))\,=\,a(\gcd(m,\,n))\,$ )
• Multiplicity sequences and Category:Multiplicity sequences ($\forall \,m,\,n\,\in \,\mathbb {N} ,\,{\rm {lcm}}(a(m),\,a(n))\,=\,a({\rm {lcm}}(m,\,n))\,$ )