A370726 -id:A370726 - cp's OEIS frontend

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

Mohammed Bouras, May 12 2024

nonn

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

			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.

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

Cf. A051403, A356360, A369797. A370726.

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

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

Original entry on oeis.org

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

Mohammed Bouras, May 12 2024

nonn

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

			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.

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

Cf. A051403, A356360, A369797. A370726.

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

cp's OEIS Frontend

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

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