# [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.

-----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
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
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

```