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?

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# Most general multiplicative sequence is ?

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