[seqfan] Re: Coprime To Sums

franktaw at netscape.net franktaw at netscape.net
Mon Aug 31 22:22:41 CEST 2009

After the initial 1,2,5, this is exactly every prime except for the 
larger of each pair of twin primes.  This is because adding two 
distinct odd primes p+q produces an even number whose largest prime 
factor must be less than max(p,q).  The same goes for 1+p, so only the 
2 actually eliminates anything.

Similar considerations apply to A164901.  If, as conjectured, the rest 
of the sequence is all primes, only the 2 and 4 effectively eliminate 
any primes -- though they do interact in a more complex fashion than 
for this sequence.

Franklin T. Adams-Watters

-----Original Message-----
From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>

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

%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 
%e A164921 The first 4 terms are 0,1,2,5. The sums of every pair of 
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 
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 
(That is because every term occurs as a sum of two distinct terms -- by 
0. And either a prime occurs in the sequence, or it doesn't because it 
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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