A169612 - OEIS

%I #45 Jul 09 2023 10:37:26

%S 1,2,4,5,8,9,10,13,14,23,24,25,30,31,32,33,38,39,40,41,46,47,48,49,54,

%T 55,60,65,70,71,72,89,90,91,92,93,94,95,120,129,146,163,164,165,182,

%U 247,264,297,298,347,396,445,446,495,544,545,578,595,612,629,646,647,648

%N Powers of 2 added sequentially to an initial value of 1 in producing A168612, which is a greedy listing of numbers all relatively prime to each other in binary.

%C The last term is a(552) = 16001. Any larger additional power of 2 added yields a number divisible by a prime no larger than 257 that is already included as a factor.

%C First computed using the online Java program in the links; the below PARI program computes all terms in about 45 seconds. - _James G. Merickel_, Aug 08 2015

%H James G. Merickel, <a href="/A169612/b169612.txt">Table of n, a(n) for n = 1..552</a>

%H Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>

%e a(5) = 8 means that with a(1)-a(4) known to be 1, 2, 4 and 5 -- corresponding to the relatively prime values 3 = 1 + 2^1, 7 = 3 + 2^2, 23 = 7 + 2^4 and 55 = 23 + 2^5 -- 55 + 2^6 = 119 has a factor of 7 and 55 + 2^7 = 183 has a factor of 3, but 55 + 2^8 = 311 is relatively prime to all of 3, 7, 23 and 55.

%o (PARI)

%o {

%o x=[1];k=1;j=1;e=2;y=1;while(1,

%o   n=e+y;f=1;for(i=1,j,

%o     if(gcd(x[i],n)!=1,f=0;break()));

%o   if(f,

%o     print1(k" ");y+=e;x=concat(x,y);j++);

%o   e+=e;k++)

%o }

%Y Cf. A168612.

%K fini,full,nonn,base

%O 1,2

%A _James G. Merickel_, Dec 03 2009

%E Name edited by _James G. Merickel_, Aug 07 2015