[seqfan] Re: Coprime To Sums

Leroy Quet q1qq2qqq3qqqq at yahoo.com
Mon Aug 31 16:20:45 CEST 2009

FYI: I just submitted the related sequence that starts with 0 and 1 instead.

%I A164921
%S A164921 0,1,2,5,11,17,23,29,37,41,47,53
%N A164921 a(1)=0, a(2)=1. For n >=3, a(n) = the smallest integer > a(n-1) that is coprime to every sum of any two distinct earlier terms of this sequence. 
%e A164921 The first 4 terms are 0,1,2,5. The sums of every pair of distinct terms are: 0+1=1, 0+2=2, 1+2=3, 0+5=5, 1+5=6, and 2+5=7. So, we are looking for the smallest integer >5 that is coprime to 1, 2, 3, 5, 6, and 7. This number, which is a(5), is 11. 
%Y A164921 A164901,A164922,A164923 
%K A164921 more,nonn
%O A164921 1,3

It is easy to see that, except for 0 and 1, all terms of this sequence are primes.
(That is because every term occurs as a sum of two distinct terms -- by adding 0. And either a prime occurs in the sequence, or it doesn't because it divides an earlier sum. Either way, every term must be comprime to all primes less that it. Therefore, every term from a(3) on is prime.)

By the way, Robert Wilson has sent Neil the b-file of A164901. So, there is no need at the moment for anyone to extend that sequence.

Thanks for the replies,
Leroy Quet

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