Additive sequences

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Additive sequences are sequences with the property

$\forall \,m,\,n\,\in \,\mathbb {N} ,\,m\,\perp \,n\,\implies \,a(m+n)\,=\,a(m)+a(n),\,$

where $\scriptstyle m\,\perp \,n\,$ means $\scriptstyle m\,$ coprime (or orthogonal) to $\scriptstyle n.\,$

Since $\scriptstyle a(n)\,=\,a(1n)\,=\,a(1)+a(n)\,$, we then have $\scriptstyle a(1)\,=\,0.\,$

See also

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