Completely additive sequences

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Completely additive sequences are sequences with the property

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

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

Examples

A001222, the "big omega" function (see Omega(n), number of prime factors of n (with multiplicity)) is a completely additive sequence. For example, $\Omega (12)=\Omega (4)+\Omega (3)$ as well as $\Omega (6)+\Omega (2)$ and $\Omega (2)+\Omega (2)+\Omega (3)$, which is 3.

Like completely multiplicative sequences, completely additive sequences are defined precisely by their values on the primes

$\Omega (p)=1,\quad p~{\rm {prime}},\,$

$\Omega (p^{n})=n,\quad p~{\rm {prime}}.\,$

Note that since 1 is the empty product of primes, we have

$\Omega (1)=0.\,$

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