A351871 -id:A351871 - cp's OEIS frontend

A355914 a(n) = gcd(b(n-1),b(n)), where b(n) = A351871(n).

1, 2, 1, 5, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 8, 2, 3, 3, 6, 1, 1, 6, 1, 1, 6, 10, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 2, 1, 1, 2, 30, 5, 5, 8, 1, 1, 4, 43, 1, 2, 1, 3, 4, 1, 3, 12, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 25, 1, 4, 1, 1, 6, 1, 1, 6

Offset: 2

N. J. A. Sloane, Sep 19 2022

nonn

In order to understand the difference between A351871 (which cycles) and A355898 (which appears to diverge), it would be helpful to understand the difference between the respective gcd sequences (this and A355899 - the latter has a very interesting graph!).

Michael S. Branicky, Table of n, a(n) for n = 2..10001

Cf. A351871, A355898, A355899.

from math import gcd
from itertools import islice
def agen():
    a = [1, 2]
    while True: g = gcd(*a); yield g; a = [a[-1], g + sum(a)//g]
print(list(islice(agen(), 85))) # Michael S. Branicky, Sep 19 2022

a(66) and beyond from Michael S. Branicky, Sep 19 2022

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)).

Original entry on oeis.org

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

N. J. A. Sloane, Sep 01 2022

nonn

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)

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.

Cf. A005711, A351871, A355899.

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);

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 *)

{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

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

A355899 The successive gcd's arising in A355898.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 3, 3, 3, 3, 3, 3, 1, 1, 3, 3, 1, 1, 1, 1, 1, 7, 17, 3, 1, 1, 3, 3, 3, 5, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 71, 71, 5, 1, 1, 5, 1, 3, 1, 13, 3, 1, 5, 15, 1, 1, 5

Offset: 1

N. J. A. Sloane, Sep 03 2022

nonn

A proof that this does not grow too rapidly would prove that A355898 diverges.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A351871, A355898, A355900, A355901.

GCD @@@ Partition[Nest[Append[#1, #3 + Total[#2]/#3] & @@ {#1, #2, GCD @@ #2} & @@ {#, #[[-2 ;; -1]], GCD[#[[-2 ;; -1]]]} &, {1, 1}, 100], 2, 1] (* Michael De Vlieger, Sep 03 2022 *)

cp's OEIS Frontend

A355914 a(n) = gcd(b(n-1),b(n)), where b(n) = A351871(n).

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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)).

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A355899 The successive gcd's arising in A355898.

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A355900 Indices of records in A355899.

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A355901 Records in A355899.

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