If [tex]a_{nm}=a_{n}a_{m}, \, \forall n,m\in\mathhbb{N} ,[/tex] then the sequence of complex terms [tex]a_{nm}[/tex] is (generally) of what form? That is, I would like to know what the most general sequence satisfying the above relation is. For example, it is clear that we must have [tex]a_{1}=1[/tex] and that for primes [tex]p_{k}[/tex] and integers [tex]\alpha_{k}[/tex] we have(adsbygoogle = window.adsbygoogle || []).push({});

[tex]a_{\prod p_{k}^{\alpha_{k}}} =\prod a_{p_{k}}^{\alpha_{k}} [/tex]

and, clearly, for any constant b, the sequence [tex]a_{k}=k^{b}[/tex] is such a sequence, what other types of sequences qualify? would the lesser requirement that [tex]a_{2n}=a_{2}a_{n}, \, \forall n\in\mathhbb{N} ,[/tex] give any more possibilities?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Most general multiplicative sequence is ?

**Physics Forums | Science Articles, Homework Help, Discussion**