OFFSET
1,2
COMMENTS
a(n) is also the number of values of any other polynomial of degree 3, except X^3.
a(n) appears to approach (2/3)*prime(n) as n increases.
REFERENCES
R. Daublebsky von Sterneck, Über die Anzahl inkongruenter Werte, die eine ganze Funktion dritten Grades annimmt, Sitzungsber. Akad. Wiss. Wien (2A) 114 (1908), 711-717.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
Thomas Brazelton, Joshua Harrington, Matthew Litman, and Tony W. H. Wong, Distinct residues of Lucas polynomials over Fp, arXiv:2103.09119 [math.NT], 2021. See p. 1.
Zhi-Hong Sun, On the theory of cubic residues and nonresidues, Acta Arithmetica 84.4 (1998): 291-335.
Zhi-Hong Sun, On the number of incongruent residues of x^4 +ax^2 +bx modulo p, Journal of Number Theory 119 (2006), 210-241. See p. 211.
FORMULA
a(n) = prime(n) - 2*floor(prime(n)/6 + 1/2), for n >= 3. - Ridouane Oudra, Jun 13 2020
for n>=3, a(n) = (2*p + (p/3))/3 with p=prime(n) and where (p/3) is the Legendre symbol. See von Sterneck, Sun, and Brazelton et al. articles. - Michel Marcus, Mar 17 2021
EXAMPLE
a(1) = 1 since the only value X^3 + X takes mod 2 is 0.
MATHEMATICA
Array[Length@ Union@ Mod[Array[#^3 + # &, #], #] &@ Prime@ # &, 62] (* Michael De Vlieger, Jan 27 2019 *)
PROG
(PARI) a(n) = #Set(vector(prime(n), k, Mod(k^3+k, prime(n)))); \\ Michel Marcus, Jan 25 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Florian Severin, Jan 24 2019
STATUS
approved