A372763 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+5))))).

13, 19, 5, 31, 37, 43, 7, 11, 61, 67, 73, 79, 17, 1, 97, 103, 109, 23, 11, 127, 1, 139, 29, 151, 157, 163, 1, 1, 181, 1, 193, 199, 41, 211, 1, 223, 229, 47, 241, 1, 1, 1, 53, 271, 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

Offset: 3

Author

Mohammed Bouras, May 12 2024

Keywords

• nonn

Comments

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

Conjecture 2: Except for 2 and 3, all primes appear in the sequence once.

Conjecture: Record values correspond to A045375(m), m > 2. - Bill McEachen, Aug 03 2024

Examples

			For n=3, 1/(2 - 3/(3 + 5)) = 8/13, so a(3)=13.
For n=4, 1/(2 - 3/(3 - 4/(4 + 5))) = 23/19, so a(4)=19.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 5)))) = 13/5, so a(5)=5.
For n=6, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6/(6 + 5))))) = 227/31, so a(6)=31.
		

Links

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

Crossrefs

Cf. A051403, A356360, A369797. A370726.

Formula

a(n) = (6n - 5)/gcd(6n - 5, A051403(n-2) + 5*A051403(n-3)).