A334919 -id:A334919 - cp's OEIS frontend

A335031 Complement of A334919.

1, 2, 3, 4, 6, 7, 10, 11, 12, 14, 15, 19, 20, 22, 24, 26, 27, 31, 32, 34, 35, 36, 39, 42, 46, 47, 50, 52, 54, 55, 59, 60, 64, 66, 67, 70, 71, 74, 75, 76, 80, 87, 90, 91, 92, 94, 99, 102, 104, 110, 111, 112, 115, 116, 119, 122, 124, 126, 127, 131, 132, 136

Offset: 1

Davide Rotondo, May 20 2020

nonn

For each term m belonging to this list, 3*m+1 or (3*m+1)/2 is a prime.

			1, 2, 3, 4, are terms because for i,j >= 1, number 3*i*j+i+j >= 5, and for i,j >= 2, number 3*i*j-i-j >= 8.

Cf. A334919.

[m:m in [1..140]|not exists(a){i:i in [1..m-1]|IsIntegral((m-i)/(1+3*i))} and not exists(b){j:j in [2..m-1]|IsIntegral((m+j)/(-1+3*j)) and (m+j)/(-1+3*j) ge 2}]; // Marius A. Burtea, Jun 04 2020

A369797 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+2))))).

Original entry on oeis.org

7, 5, 13, 2, 19, 11, 5, 1, 31, 17, 37, 1, 43, 23, 1, 1, 1, 29, 61, 1, 67, 1, 73, 1, 79, 41, 1, 1, 1, 47, 97, 1, 103, 53, 109, 1, 1, 59, 1, 1, 127, 1, 1, 1, 139, 71, 1, 1, 151, 1, 157, 1, 163, 83, 1, 1, 1, 89, 181, 1, 1, 1, 193, 1, 199, 101, 1, 1, 211

Offset: 3

Mohammed Bouras, Feb 25 2024

nonn

Conjecture: The sequence contains only 1's and the primes.
Conjecture: The sequence of record values is A002476. - Bill McEachen, Mar 24 2024
a(n) = 1 positions appear to correspond to A334919(m) - 1, m > 2. - Bill McEachen, Aug 05 2024

			For n=3, 1/(2 - 3/(3 + 2)) = 5/7, so a(3)=7.
For n=4, 1/(2 - 3/(3 - 4/(4 + 2))) = 7/5, so a(4)=5.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 2)))) = 41/13, so a(5)=13.

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

Cf. A051403, A356360.

from math import gcd, factorial
def A369797(n): return (a:=3*n-2)//gcd(a,a*sum(factorial(k) for k in range(n-2))+n*factorial(n-2)>>1) # Chai Wah Wu, Feb 26 2024

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

cp's OEIS Frontend

A335031 Complement of A334919.

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