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)).

1, 1, 3, 5, 9, 15, 11, 27, 39, 25, 65, 23, 89, 113, 203, 317, 521, 839, 1361, 2201, 3563, 5765, 9329, 15095, 24425, 7909, 32335, 40245, 14521, 54767, 69289, 124057, 193347, 317405, 46443, 363849, 136767, 166875, 101217, 89367, 63531, 50969, 114501, 165471, 93327, 86269, 179597, 265867, 445465, 711333

Offset: 1

Author

N. J. A. Sloane, Sep 01 2022

Keywords

• nonn

Comments

Suggested by A351871.

Sequence appears to diverge, but it would be nice to have a proof.

From Giorgos Kalogeropoulos, Nov 01 2022 : (Start)

Conjecture: For n >= 3775 a(n) can also be expressed in the following three ways:

1) a(n) = 1 + a(n-1) + a(n-2).

2) a(n) = 2*a(n-1) - a(n-3).

3) If A = a(3774), B = a(3772) and F = Fibonacci A000045(n),

a(n) = (A+1)*F(n-3772) - (B+1)*F(n-3774) - 1.

These three formulas only work for n >= 3775. (End)

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..5000 (terms 1..1002 from N. J. A. Sloane)
• Michael De Vlieger, Labeled scalar plot of m = log_2(a(n)), 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.
• Peter Munn, Logarithmic plot of a(n)/A005711(n). (Note that A005711 has essentially constant exponential growth.)
• Mathematics Stack Exchange user Augusto Santi, A singular variant of the OEIS sequence A349576.
• 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: Log plot 5000 terms, Log plot 10000 terms, Log plot 100000 terms.

Crossrefs

Cf. A005711, A351871, A355899.

Programs

Maple

A351871 := proc(u,v,M) local n,r,s,g,t,a;
a:=[u,v]; r:=u; s:=v;
for n from 1 to M do g:=gcd(r,s); t:=g+(r+s)/g; a:=[op(a),t];
   r:=s; s:=t; od;
a;
end proc;
A351871(1,1,100);

Mathematica

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 *)

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

Python

from math import gcd
from itertools import islice
def A355898_gen(): # generator of terms
    yield from (a:=(1,1))
    while True: yield (a:=(a[1],(b:=gcd(*a))+sum(a)//b))[1]
A355898_list = list(islice(A355898_gen(),30)) # Chai Wah Wu, Sep 01 2022