OFFSET
3,1
COMMENTS
Conjecture: Except for 49, every term of this sequence is either a prime or 1.
LINKS
Mohammed Bouras, The Distribution Of Prime Numbers And Continued Fractions, (ppt) (2022)
FORMULA
Except for n=6, if gpf(n^2 + 3*n - 5) > n, then we have:
a(n) = gpf(n^2 + 3*n - 5), where gpf = "greatest prime factor".
If a(n) = a(m) and n < m < a(n), then we have:
a(n) = n + m + 3.
a(n) divides gcd(n^2 + 3*n - 5, m^2 + 3*m - 5).
EXAMPLE
For n=3, 1/(2 - 3/(-5)) = 5/13, so a(3) = 13.
For n=4, 1/(2 - 3/(3 - 4/(-5))) = 19/23, so a(4) = 23.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(-5)))) = 11/7, so a(5) = 7.
PROG
(PARI) lf(n) = sum(k=0, n-1, k!); \\ A003422
f(n) = (n+2)*lf(n+1)/2; \\ A051403
a(n) = (n^2 + 3*n - 5)/gcd(n^2 + 3*n - 5, 5*f(n-3) + n*f(n-4)); \\ Michel Marcus, Jun 06 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Mohammed Bouras, Jun 04 2023
STATUS
approved