A370726 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+3))))).

3, 13, 17, 7, 5, 29, 11, 37, 41, 1, 7, 53, 19, 61, 1, 23, 73, 1, 1, 1, 89, 31, 97, 101, 1, 109, 113, 1, 1, 1, 43, 1, 137, 47, 1, 149, 1, 157, 1, 1, 1, 173, 59, 181, 1, 1, 193, 197, 67, 1, 1, 71, 1, 1, 1, 229, 233, 79, 241, 1, 83, 1, 257, 1, 1, 269, 1, 277

Offset: 3

Author

Mohammed Bouras, Feb 28 2024

Keywords

• nonn

Comments

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

Conjecture: Record values correspond to A002144 (n>3). - Bill McEachen, May 21 2024

Examples

			For n=3, 1/(2 - 3/(3 + 3)) = 2/3, so a(3)=3.
For n=4, 1/(2 - 3/(3 - 4/(4 + 3))) = 17/13, so a(4)=13.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 3)))) = 49/17, so a(5)=17.
		

Links

• Bill McEachen, Table of n, a(n) for n = 3..10002
• Mohammed Bouras, The Distribution Of Prime Numbers And The Continued Fractions, (paper still under development) (2022).
• Mohammed Bouras, The Distribution Of Prime Numbers And Continued Fractions, (ppt) (2022).

Crossrefs

Cf. A051403, A356360, A369797.

Formula

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