A370726 - OEIS

%I #18 May 21 2024 11:30:15

%S 3,13,17,7,5,29,11,37,41,1,7,53,19,61,1,23,73,1,1,1,89,31,97,101,1,

%T 109,113,1,1,1,43,1,137,47,1,149,1,157,1,1,1,173,59,181,1,1,193,197,

%U 67,1,1,71,1,1,1,229,233,79,241,1,83,1,257,1,1,269,1,277

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

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

%C Conjecture: Record values correspond to A002144 (n>3). - _Bill McEachen_, May 21 2024

%H Bill McEachen, <a href="/A370726/b370726.txt">Table of n, a(n) for n = 3..10002</a>

%H Mohammed Bouras, <a href="https://doi.org/10.5281/zenodo.7212512">The Distribution Of Prime Numbers And The Continued Fractions</a>, (paper still under development) (2022).

%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) = (4n - 3)/gcd(4n - 3, A051403(n-2) + 3*A051403(n-3)).

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

%e For n=4, 1/(2 - 3/(3 - 4/(4 + 3))) = 17/13, so a(4)=13.

%e For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 3)))) = 49/17, so a(5)=17.

%Y Cf. A051403, A356360, A369797.

%K nonn

%O 3,1

%A _Mohammed Bouras_, Feb 28 2024