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a(0) = 0; a(1) = 1; for n >= 2, a(n) = the n-th integer from among those positive integers which are coprime to a(n-1)+a(n-2).
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%I #17 Oct 11 2019 11:30:27

%S 0,1,2,4,11,8,6,15,13,19,19,23,41,25,43,31,31,35,59,37,59,61,79,67,47,

%T 77,53,71,57,57,91,63,81,97,67,71,109,137,115,137,139,127,103,111,87,

%U 149,93,103,111,97,107,161,105,129,175,115,143,173,117,151,121,129,153

%N a(0) = 0; a(1) = 1; for n >= 2, a(n) = the n-th integer from among those positive integers which are coprime to a(n-1)+a(n-2).

%C For n>=9, a(n-1)+a(n-2) is even so a(n) is odd and >= 2n-1. It appears that a(n)=2n-1 infinitely often, when a(n-1)+a(n-2)=2p with p prime. A good upper bound is not known; presumably a(n)/n is unbounded, since it's large whenever a(n-1)+a(n-2) is divisible by many small primes. - _Dean Hickerson_, Dec 07 2006

%C Zak Seidov observed that straight lines are visible in the graph of a(n). The lowest line is given by the points with a(n)=2n-1. There are also many points a short distance above this line; these occur when a(n-1)+a(n-2) is a small power of 2 times a prime. Other lines show points with a(n-1)+a(n-2) equal to a small number times a large prime; e.g. when n>18 and a(n-1)+a(n-2)=6p, a(n) is either 3n+1 or 3n+2. - _Dean Hickerson_, Dec 07 2006

%H Zak Seidov, <a href="/A124183/b124183.txt">Table of n, a(n) for n = 0..999.</a>

%e a(6)+a(7)=21. 1,2,4,5,8,10,11,13,16,17,... are the positive integers which are coprime to 21. 13 is the 8th of these integers, so a(8) = 13.

%t f[l_List] := Block[{k = 0, c = Length[l]},While[c > 0,k++;While[GCD[k, l[[ -1]] + l[[ -2]]] > 1, k++ ];c--;];Append[l, k]];Nest[f, {0, 1}, 65] (* _Ray Chandler_, Dec 06 2006 *)

%t a[0]=0; a[1]=1; a[n_]:=a[n]=Module[{s,c,v}, s=a[n-1]+a[n-2]; v=0; For[c=1, c<=n, c++, While[GCD[ ++v,s]>1, Null]]; v] (* _Hickerson_* )

%Y Cf. A121047.

%K nonn

%O 0,3

%A _Leroy Quet_, Dec 05 2006

%E Extended by _Ray Chandler_ and _Zak Seidov_, Dec 06 2006