A372761 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+4))))).

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, 47, 73, 151, 1, 1, 83, 1, 1, 181, 1, 191, 1, 67, 103, 211, 1, 1, 113, 1, 59, 241, 1, 251, 1, 1, 1, 271, 1, 281, 1, 97, 1, 1, 1, 311, 79, 107, 163, 331, 1, 1, 173, 1

Offset: 3

Author

Mohammed Bouras, May 12 2024

Keywords

• nonn

Comments

Conjecture 1: Except for 4, the sequence contains only 1's and the primes.

Conjecture 2: Except for 3 and 5, all odd primes appear in the sequence once.

Conjecture: Record values correspond to A030430 (except a(6) = 13). - Bill McEachen, Aug 03 2024

Examples

			For n=3, 1/(2 - 3/(3 + 4)) = 7/11, so a(3)=11.
For n=4, 1/(2 - 3/(3 - 4/(4 + 4))) = 5/4, so a(4)=4.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 4)))) = 19/7, so a(5)=7.
For n=6, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6/(6 + 4))))) = 101/13, so a(6)=13.
		

Links

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

Crossrefs

Cf. A051403, A356360, A369797. A370726.

Formula

a(n) = (5n - 4)/gcd(5n - 4, A051403(n-2) + 4*A051403(n-3)).