"logarithmetic" sequences

Gerald McGarvey Gerald.McGarvey at comcast.net
Thu Apr 14 20:04:56 CEST 2005

```Here are some relevant links:

http://mathworld.wolfram.com/SumofPrimeFactors.html
http://mathworld.wolfram.com/RhondaNumber.html
http://mathworld.wolfram.com/UnorderedFactorization.html
(A086436 and A001222 are the same except for the first term.)

Gerald

At 04:24 PM 4/12/2005, Marc LeBrun wrote:
>Speaking of multiplicative sequences, ie those that (perhaps with
>conditions) obey
>
>   a(mn) = a(m) a(n)
>
>I was wondering what might be said for sequences that instead satisfy
>
>   a(mn) = a(m) + a(n)
>
>which might be called "log-arithmetic" (or "logarithm-etic" if you prefer)
>by analogy with regular logs.
>
>Have these been discussed here before?  Is there any standard name for them?
>
>Of course, as with their multiplicative cousins, logarithmetic sequences
>can be generated by their prime-power index subsequence.
>
>So by summing f(p^e) for various easy f we get A001221, A001222, A056169,
>A001414, and A008472 for instance.
>
>Generalizing + to g(.) we can further posit "factorization-defined" sequences
>
>   a(n) = g(f(p1^e1),f(p2^e2),...)
>
>for various f & g, and even imagine superseekeresque detection of
>them.  (And even more generally, to use other unique expansions).
>
>More practically, if someone has the wherewithal to crunch the database,
>it might be interesting to simply collect all the existing instances of
>candidate logarithmetic sequences, since there seem to be quite a few.
>
>It might be even more immediately useful to check for any entries
>potentially missing their "mult" keywords.
>
>(Just to guiltily throw a few more suggestions onto the endless pile!)
>

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