A369797 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+2))))).

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

Offset: 3

Author

Mohammed Bouras, Feb 25 2024

Keywords

• nonn

Comments

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

Conjecture: The sequence of record values is A002476. - Bill McEachen, Mar 24 2024

a(n) = 1 positions appear to correspond to A334919(m) - 1, m > 2. - Bill McEachen, Aug 05 2024

Examples

			For n=3, 1/(2 - 3/(3 + 2)) = 5/7, so a(3)=7.
For n=4, 1/(2 - 3/(3 - 4/(4 + 2))) = 7/5, so a(4)=5.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 2)))) = 41/13, so a(5)=13.
		

Links

• Mohammed Bouras, The Distribution Of Prime Numbers And Continued Fractions, (ppt) (2022)

Crossrefs

Cf. A051403, A356360.

Programs

Python

from math import gcd, factorial
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

Formula

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