A356360 - OEIS

%I #31 Aug 06 2024 22:00:18

%S 5,7,3,11,13,1,17,19,1,23,1,1,29,31,1,1,37,1,41,43,1,47,1,1,53,1,1,59,

%T 61,1,1,67,1,71,73,1,1,79,1,83,1,1,89,1,1,1,97,1,101,103,1,107,109,1,

%U 113,1,1,1,1,1,1,127,1,131,1,1,137,139,1,1,1,1,149,151,1,1,157,1,1,163,1,167

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

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

%C Similar continued fraction to A356247.

%C Same as A128059(n), A145737(n-1) and A097302(n-2) for n > 5.

%C a(n) = 1 positions appear to correspond to A104275(m), m > 2. Conjecture: all odd primes are seen in order after 11. - _Bill McEachen_, Aug 05 2024

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

%F For n >= 3, the formula of the continued fraction is as follows:

%F (A051403(n-2) + A051403(n-3))/(2n - 1) = 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).

%F a(n) = (2n - 1)/gcd(2n - 1, A051403(n-2) + A051403(n-3)).

%F From the conjecture: Except for n = 5, a(n)= 2n - 1 if 2n-1 is prime, 1 otherwise.

%o (Python)

%o from math import gcd, factorial

%o def A356360(n): return (a:=(n<<1)-1)//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, A051417, A097302, A128059, A145737, A356247.

%Y Cf. A104275.

%K nonn

%O 3,1

%A _Mohammed Bouras_, Oct 15 2022