A356360 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).

5, 7, 3, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1, 1, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 1, 1, 1, 127, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1, 149, 151, 1, 1, 157, 1, 1, 163, 1, 167

Offset

3,1

Comments

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

Similar continued fraction to A356247.

Same as A128059(n), A145737(n-1) and A097302(n-2) for n > 5.

a(n) = 1 positions appear to correspond to A104275(m), m > 2. Conjecture: all odd primes are seen in order after 11. - Bill McEachen, Aug 05 2024

Links

Table of n, a(n) for n=3..84.

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

Formula

For n >= 3, the formula of the continued fraction is as follows:

(A051403(n-2) + A051403(n-3))/(2n - 1) = 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).

a(n) = (2n - 1)/gcd(2n - 1, A051403(n-2) + A051403(n-3)).

From the conjecture: Except for n = 5, a(n)= 2n - 1 if 2n-1 is prime, 1 otherwise.

Prog

(Python)
from math import gcd, factorial
def A356360(n): return (a:=(n<<1)-1)//gcd(a, a*sum(factorial(k) for k in range(n-2))+n*factorial(n-2)>>1) # Chai Wah Wu, Feb 26 2024

Crossrefs

Cf. A051403, A051417, A097302, A128059, A145737, A356247.

Cf. A104275.

Sequence in context: A151753 A144543 A205135 * A076567 A146535 A141650

Adjacent sequences: A356357 A356358 A356359 * A356361 A356362 A356363

Keyword

nonn

Author

Mohammed Bouras, Oct 15 2022

Status

approved