Talk:Completely additive sequences

Examples

What's a good example of one of these? Besides A000027? Alonso del Arte 01:55, 24 July 2012 (UTC)

A001222 Omega(n), number of prime factors of n (with multiplicity)

A064415 a(1) = 0, a(n) = iter(n) if n is even, a(n) = iter(n)-1 if n is odd, where iter(n) = A003434(n) = smallest number of iterations of Euler totient function phi needed to reach 1.

A076649 Number of digits required to write the prime factors of n.

Obtained by searching the OEIS for "completely additive" with the quotes. There seems to be very few of them... — Daniel Forgues 05:13, 24 July 2012 (UTC)

A001222 and A001414 are probably the most interesting. A000004, (aside from offset) A001478, and A001489 are trivial examples. Charles R Greathouse IV 07:04, 24 July 2012 (UTC)

Are you sure big omega is completely additive? How do you deal with 1? What about if m = n, in the case of even numbers? Alonso del Arte 01:38, 25 July 2012 (UTC)

Omega(n), number of prime factors of n (with multiplicity) is completely additive, since

$m=\prod _{i=1}^{\omega (m)}{p_{i}}^{\alpha _{i}},\,n=\prod _{i=1}^{\omega (n)}{q_{i}}^{\beta _{i}},\,$

where $\scriptstyle \omega (n)\,$ is the number of distinct prime factors of n, gives

$\Omega (m)=\sum _{i=1}^{\omega (m)}{\alpha _{i}},\,\Omega (n)=\sum _{i=1}^{\omega (n)}{\beta _{i}},\,$

we then have

$mn=\left(\prod _{i=1}^{\omega (m)}{p_{i}}^{\alpha _{i}}\right)\,\left(\prod _{i=1}^{\omega (n)}{q_{i}}^{\beta _{i}}\right),\,$

giving

$\Omega (mn)=\sum _{i=1}^{\omega (m)}{\alpha _{i}}+\sum _{i=1}^{\omega (n)}{\beta _{i}}=\Omega (m)+\Omega (n).\,$

Since we have "with multiplicity," it doesn't matter what $\scriptstyle m\,$ and $\scriptstyle n\,$ are.

Since 1 is the empty product of primes, $\scriptstyle \Omega (1)\,=\,0\,$ (1 has 0 prime factors).

$\Omega (n)=\Omega (1n)=\Omega (1)+\Omega (n)=0+\Omega (n)=\Omega (n).\,$ — Daniel Forgues 04:34, 25 July 2012 (UTC)

That's not what I understood from what the page says. I tried Omega(12) = Omega(1) + Omega(11) and that's wrong. Alonso del Arte 01:49, 26 July 2012 (UTC)

I'll check if the page is incorrect or misleading. But what Daniel writes above is correct, Omega is completely additive (and the usual example of a completely additive function). For 12, Omega(12) = Omega(4) + Omega(3) = Omega(2) + Omega(2) + Omega(3) = Omega(6) + Omega(2) = 3. Charles R Greathouse IV 06:31, 26 July 2012 (UTC)

You are quite right, Alonso, in pointing out that error. I've corrected it on this page and on {{Classification of sequences by mathematical property}}. Thanks! Charles R Greathouse IV 06:35, 26 July 2012 (UTC)

Thanks, Charles for clearing that up. Of course this means that the first example I thought of, A27, is in fact not an additive sequence at all. Alonso del Arte 11:36, 26 July 2012 (UTC)

Right. (That other definition is rather uninteresting, since it consists precisely of multiples of A000027.) Charles R Greathouse IV 18:40, 26 July 2012 (UTC)

Oops! Sorry about the typo. (A000027 is trivially completely multiplicative...) — Daniel Forgues 07:00, 29 July 2012 (UTC)