[seqfan] Re: Coprime To Sums

David Wilson dwilson at gambitcomm.com
Mon Aug 31 16:20:13 CEST 2009

While we're on the topic of coprime sums...

A062096 is a very interesting sequence.

I came up with A062096 at one time early in my work on the OEIS, which 
would have been in the A02 or A03 days, but I never published. It's a 
simple idea, and it looks like the prolific Amarnath Murthy beat me to 
submission (though as was often the case, his submission had to be later 
corrected). I'm guessing Murthy did not submit analogous sequences with 
larger starting elements.

At any rate, the idea is simple. Let a_k(1) = k, and thereafter, choose 
a(n+1) to be the smallest number > a(n) coprime to the sum of a(1) 
through a(n).

For k = 2, the sequence looks randomish, with messy blocks punctuated by 
periodic blocks, but apparently never settling on a permanent period.

For k = 3, the sequence is 3, followed by larger numbers not divisible 
by 3, hence periodic. This is easily proved.

For k = 4, the sequence is 4 followed by the larger odd numbers, hence 
periodic. This is easily proved.

For the next few k >= 5, the sequence again appears randomish as for k = 2.

Anyway, the sequences are interesting to study.

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