A323703 Number of values of (X^3 + X) mod prime(n).

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

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

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

Adjacent sequences: A323700 A323701 A323702 * A323704 A323705 A323706

Keyword

nonn

Author

Florian Severin, Jan 24 2019

Status

approved