Additive sequences

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Additive sequences are sequences with the property

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

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

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

See also

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

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

• Coprimality sequences and Category:Coprimality sequences ($\mathrm{\forall }m,n\in \mathbb{\mathbb{N}},m\perp n⟹a(m)\perp a(n)$)
  • Coprime sequences and Category:Coprime sequences ($\mathrm{\forall }m,n\in \mathbb{\mathbb{N}},m\ne n⟹a(m)\perp a(n)$)

• Divisibility sequences and Category:Divisibility sequences ($\mathrm{\forall }m,n\in \mathbb{\mathbb{N}},m|n⟹a(m)|a(n)$)
• Strong divisibility sequences and Category:Strong divisibility sequences ($\mathrm{\forall }m,n\in \mathbb{\mathbb{N}},\gcd (a(m),a(n))=a(\gcd (m,n))$)
• Multiplicity sequences and Category:Multiplicity sequences ($\mathrm{\forall }m,n\in \mathbb{\mathbb{N}},\mathrm{l}\mathrm{c}\mathrm{m}(a(m),a(n))=a(\mathrm{l}\mathrm{c}\mathrm{m}(m,n))$)