This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372763 #8 Aug 03 2024 19:04:18 %S A372763 13,19,5,31,37,43,7,11,61,67,73,79,17,1,97,103,109,23,11,127,1,139,29, %T A372763 151,157,163,1,1,181,1,193,199,41,211,1,223,229,47,241,1,1,1,53,271, %U A372763 277,283,1,59,1,307,313,1,1,331,337,1,349,71,1,367,373,379,1,1,397,1,409,83,421 %N A372763 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+5))))). %C A372763 Conjecture 1: The sequence contains only 1's and the primes. %C A372763 Conjecture 2: Except for 2 and 3, all primes appear in the sequence once. %C A372763 Conjecture: Record values correspond to A045375(m), m > 2. - _Bill McEachen_, Aug 03 2024 %H A372763 Mohammed Bouras, <a href="https://doi.org/10.5281/zenodo.10992128">The Distribution Of Prime Numbers And Continued Fractions</a>, (ppt) (2022). %F A372763 a(n) = (6n - 5)/gcd(6n - 5, A051403(n-2) + 5*A051403(n-3)). %e A372763 For n=3, 1/(2 - 3/(3 + 5)) = 8/13, so a(3)=13. %e A372763 For n=4, 1/(2 - 3/(3 - 4/(4 + 5))) = 23/19, so a(4)=19. %e A372763 For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 5)))) = 13/5, so a(5)=5. %e A372763 For n=6, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6/(6 + 5))))) = 227/31, so a(6)=31. %Y A372763 Cf. A051403, A356360, A369797. A370726. %K A372763 nonn %O A372763 3,1 %A A372763 _Mohammed Bouras_, May 12 2024