Starting with $n=0$:
$k=2$
Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers,
$$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$
The $n$ that divides $A_n-1$ are all the primes and the Lucas pseudoprimes,
$$n = 705, 2465, 2737, 3745, 4181,\dots$$
$k=3$
Given the roots $x_i$ of $x^3-x^2-x-1=0$. Then,
$$B_n = x_1^n+x_2^n+x_3^n = 3,1,3,7,11,21,39,\dots$$
The tribonacci-like pseudoprimes $n$ that divide $B_n-1$ are,
$$n = 182,25201,\color{brown}{233^2},63618,194390,750890,804055,\dots,\color{brown}{233^3},\dots$$
See this post for a list by S. Stadnicki.
$k=4$
Given the roots $x_i$ of $x^4-x^3-x^2-x-1=0$. Then,
$$C_n = x_1^n+x_2^n+x_3^n+x_4^n = 4, 1, 3, 7, 15, 26, 51, 99,\dots$$
given by (A073817). The tetranacci-like pseudoprimes $n$ that divide $C_n-1$ are,
$$n = 5^2,\,7^2,\,5^3,\,13^2,\,7^3,\,493,\,37^2,7825,\,29877\dots$$
and for $k=4$, these are all for $n<50000$.
Questions:
- What are the tetranacci-like pseudoprimes for $n<10^6$ or higher? (And which of them are squares or cubes?)
- Why are the small tetranacci-like pseudoprimes powers of primes? (In contrast to the Lucas or tribonacci-like pseudoprimes.)
