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A323703
Number of values of (X^3 + X) mod prime(n).
1
1, 3, 3, 5, 7, 9, 11, 13, 15, 19, 21, 25, 27, 29, 31, 35, 39, 41, 45, 47, 49, 53, 55, 59, 65, 67, 69, 71, 73, 75, 85, 87, 91, 93, 99, 101, 105, 109, 111, 115, 119, 121, 127, 129, 131, 133, 141, 149, 151, 153, 155, 159, 161, 167, 171, 175, 179, 181, 185, 187, 189, 195
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
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
Cf. A323704 (the number of values of X^3), A130291 (the number of values of X^2, which is also the number of values of any other polynomial of degree 2).
Sequence in context: A247130 A271974 A050824 * A333147 A323434 A323431
KEYWORD
nonn
AUTHOR
Florian Severin, Jan 24 2019
STATUS
approved