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A355898 a(1) = a(2) = 1; a(n) = gcd(a(n-1), a(n-2)) + (a(n-1) + a(n-2))/gcd(a(n-1), a(n-2)).

%I A355898 #46 Nov 02 2022 07:53:56
%S A355898 1,1,3,5,9,15,11,27,39,25,65,23,89,113,203,317,521,839,1361,2201,3563,
%T A355898 5765,9329,15095,24425,7909,32335,40245,14521,54767,69289,124057,
%U A355898 193347,317405,46443,363849,136767,166875,101217,89367,63531,50969,114501,165471,93327,86269,179597,265867,445465,711333
%N A355898 a(1) = a(2) = 1; a(n) = gcd(a(n-1), a(n-2)) + (a(n-1) + a(n-2))/gcd(a(n-1), a(n-2)).
%C A355898 Suggested by A351871.
%C A355898 Sequence appears to diverge, but it would be nice to have a proof.
%C A355898 From _Giorgos Kalogeropoulos_, Nov 01 2022 : (Start)
%C A355898 Conjecture: For n >= 3775 a(n) can also be expressed in the following three ways:
%C A355898 1) a(n) = 1 + a(n-1) + a(n-2).
%C A355898 2) a(n) = 2*a(n-1) - a(n-3).
%C A355898 3) If A = a(3774), B = a(3772) and F = Fibonacci A000045(n),
%C A355898    a(n) = (A+1)*F(n-3772) - (B+1)*F(n-3774) - 1.
%C A355898 These three formulas only work for n >= 3775. (End)
%H A355898 Seiichi Manyama, <a href="/A355898/b355898.txt">Table of n, a(n) for n = 1..5000</a> (terms 1..1002 from N. J. A. Sloane)
%H A355898 Michael De Vlieger, <a href="/A355898/a355898.png">Labeled scalar plot of m = log_2(a(n))</a>, n = 1..2^12, highlighting areas with near-zero second differences of log_2(a(n)) in red, otherwise blue. Labels are indices that begin and end a run of second differences near zero. The third run begins at n approximately 3797 but continues at least to n = 2^16. "Near-zero" means m > 10^-10.
%H A355898 Peter Munn, <a href="https://oeis.org/plot2a?name1=A355898&amp;name2=A005711&amp;tform1=log+base+10&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Logarithmic plot of a(n)/A005711(n)</a>. (Note that A005711 has essentially constant exponential growth.)
%H A355898 Mathematics Stack Exchange user Augusto Santi, <a href="https://math.stackexchange.com/questions/4387881/a-singular-variant-of-the-oeis-sequence-a349576">A singular variant of the OEIS sequence A349576</a>.
%H A355898 Giorgos Kalogeropoulos, After the term a(3773) it appears that the logarithmic graph is a straight line. This happens because the GCD of two successive terms from a(3773) and on is equal to 1. I tested all the terms up to a(10^6). If this holds to infinity then the sequence diverges. Here are the log graphs:  <a href="/A355898/a355898_1.png">Log plot 5000 terms</a>, <a href="/A355898/a355898_2.png">Log plot 10000 terms</a>, <a href="/A355898/a355898_3.png">Log plot 100000 terms</a>.
%p A355898 A351871 := proc(u,v,M) local n,r,s,g,t,a;
%p A355898 a:=[u,v]; r:=u; s:=v;
%p A355898 for n from 1 to M do g:=gcd(r,s); t:=g+(r+s)/g; a:=[op(a),t];
%p A355898    r:=s; s:=t; od;
%p A355898 a;
%p A355898 end proc;
%p A355898 A351871(1,1,100);
%t A355898 Nest[Append[#1, #3 + Total[#2]/#3] & @@ {#1, #2, GCD @@ #2} & @@ {#, #[[-2 ;; -1]], GCD[#[[-2 ;; -1]]]} &, {1, 1}, 48] (* _Michael De Vlieger_, Sep 03 2022 *)
%o A355898 (Python)
%o A355898 from math import gcd
%o A355898 from itertools import islice
%o A355898 def A355898_gen(): # generator of terms
%o A355898     yield from (a:=(1,1))
%o A355898     while True: yield (a:=(a[1],(b:=gcd(*a))+sum(a)//b))[1]
%o A355898 A355898_list = list(islice(A355898_gen(),30)) # _Chai Wah Wu_, Sep 01 2022
%o A355898 (PARI) {a355898(N=50,A1=1,A2=1)= my(a=vector(N));a[1]=A1;a[2]=A2;for(n=1,N,if(n>2,my(g=gcd(a[n-1],a[n-2]));a[n]=g+(a[n-1]+a[n-2])/g);print1(a[n],",")) } \\ _Ruud H.G. van Tol_, Sep 19 2022
%Y A355898 Cf. A005711, A351871, A355899.
%K A355898 nonn
%O A355898 1,3
%A A355898 _N. J. A. Sloane_, Sep 01 2022