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 A372761 #7 Aug 03 2024 19:03:59 %S A372761 11,4,7,13,31,1,41,23,17,1,61,1,71,19,1,43,1,1,101,53,37,29,1,1,131,1, %T A372761 47,73,151,1,1,83,1,1,181,1,191,1,67,103,211,1,1,113,1,59,241,1,251,1, %U A372761 1,1,271,1,281,1,97,1,1,1,311,79,107,163,331,1,1,173,1 %N A372761 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+4))))). %C A372761 Conjecture 1: Except for 4, the sequence contains only 1's and the primes. %C A372761 Conjecture 2: Except for 3 and 5, all odd primes appear in the sequence once. %C A372761 Conjecture: Record values correspond to A030430 (except a(6) = 13). - _Bill McEachen_, Aug 03 2024 %H A372761 Mohammed Bouras, <a href="https://doi.org/10.5281/zenodo.10992128">The Distribution Of Prime Numbers And Continued Fractions</a>, (ppt) (2022). %F A372761 a(n) = (5n - 4)/gcd(5n - 4, A051403(n-2) + 4*A051403(n-3)). %e A372761 For n=3, 1/(2 - 3/(3 + 4)) = 7/11, so a(3)=11. %e A372761 For n=4, 1/(2 - 3/(3 - 4/(4 + 4))) = 5/4, so a(4)=4. %e A372761 For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 4)))) = 19/7, so a(5)=7. %e A372761 For n=6, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6/(6 + 4))))) = 101/13, so a(6)=13. %Y A372761 Cf. A051403, A356360, A369797. A370726. %K A372761 nonn %O A372761 3,1 %A A372761 _Mohammed Bouras_, May 12 2024