A363482 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(-5))))).
13, 23, 7, 49, 13, 83, 103, 5, 149, 1, 29, 233, 53, 23, 67, 373, 59, 1, 499, 109, 593, 643, 139, 107, 1, 863, 71, 197, 1049, 223, 1, 179, 53, 1399, 59, 1553, 71, 1, 257, 1, 1973, 2063, 431, 173, 67, 349, 2543, 1, 2749, 571, 2963, 439, 1, 3299, 683, 3533, 281, 151, 557, 1, 4153
Offset: 3
Keywords
Examples
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.
Links
- Bill McEachen, Table of n, a(n) for n = 3..10000
- Mohammed Bouras, The Distribution Of Prime Numbers And Continued Fractions, (ppt) (2022)
Programs
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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
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).
Comments