A369797 - OEIS

%I #33 Aug 06 2024 22:00:40

%S 7,5,13,2,19,11,5,1,31,17,37,1,43,23,1,1,1,29,61,1,67,1,73,1,79,41,1,

%T 1,1,47,97,1,103,53,109,1,1,59,1,1,127,1,1,1,139,71,1,1,151,1,157,1,

%U 163,83,1,1,1,89,181,1,1,1,193,1,199,101,1,1,211

%N Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+2))))).

%C Conjecture: The sequence contains only 1's and the primes.

%C Conjecture: The sequence of record values is A002476. - _Bill McEachen_, Mar 24 2024

%C a(n) = 1 positions appear to correspond to A334919(m) - 1, m > 2. - _Bill McEachen_, Aug 05 2024

%H Mohammed Bouras, <a href="https://doi.org/10.5281/zenodo.10992128">The Distribution Of Prime Numbers And Continued Fractions</a>, (ppt) (2022)

%F a(n) = (3n - 2)/gcd(3n - 2, A051403(n-2) + 2*A051403(n-3)).

%e For n=3, 1/(2 - 3/(3 + 2)) = 5/7, so a(3)=7.

%e For n=4, 1/(2 - 3/(3 - 4/(4 + 2))) = 7/5, so a(4)=5.

%e For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 2)))) = 41/13, so a(5)=13.

%o (Python)

%o from math import gcd, factorial

%o def A369797(n): return (a:=3*n-2)//gcd(a,a*sum(factorial(k) for k in range(n-2))+n*factorial(n-2)>>1) # _Chai Wah Wu_, Feb 26 2024

%Y Cf. A051403, A356360.

%K nonn

%O 3,1

%A _Mohammed Bouras_, Feb 25 2024