Talk:Coprimality sequences

Would Sylvester's sequence (A000058) count as an example of a coprimality sequence? Alonso del Arte 00:01, 30 July 2012 (UTC)

Trivially so, as are all sequences where gcd(a(m), a(n)) = 1 for all m and n. Charles R Greathouse IV 01:14, 30 July 2012 (UTC)

I should comment, though, that I've never read anything about these, not even a mention, in the literature. Charles R Greathouse IV 01:15, 30 July 2012 (UTC)

This is the first time I've come across the term, too.

If this is really a thing, I find them more interesting if they contain composite numbers, or maybe even only composite numbers. Alonso del Arte 03:03, 30 July 2012 (UTC)

I believe that the relative density of primes in Sylvester's sequence is 0, so although it doesn't consist of only primes the primes are rare enough that the condition seems meaningful. But I still find the example trivial because all gcds are 1, not just those required by the condition. Charles R Greathouse IV 03:58, 30 July 2012 (UTC)

Coprime sequences (like the Fermat numbers) are obviously trivially coprimality sequences... I don't know what would be interesting examples of coprimality sequences which are not coprime sequences... — Daniel Forgues 07:56, 30 July 2012 (UTC)

Since for the sequence of Fibonacci numbers, we have $\scriptstyle \gcd(F_{m},F_{n})\,=\,F(\gcd(m,n))\,$ (a strong divisibility sequence), this is a coprimality sequence since $\scriptstyle F(1)\,=\,1\,$, although not a coprime sequence ($\scriptstyle \gcd(F_{m},F_{n})\,=\,1\,$ also for $\scriptstyle \gcd(m,n)\,=\,2\,$ since $\scriptstyle F(2)\,=\,1\,$)! The sequence of Fibonacci numbers is also a divisibility sequence. — Daniel Forgues 08:33, 30 July 2012 (UTC)

Now, it would be interesting to find a coprimality sequence (but not a coprime sequence) which is not a divisibility sequence... — Daniel Forgues 08:33, 30 July 2012 (UTC)