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

%I A323703 #67 Feb 23 2022 07:13:04
%S A323703 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,
%T A323703 67,69,71,73,75,85,87,91,93,99,101,105,109,111,115,119,121,127,129,
%U A323703 131,133,141,149,151,153,155,159,161,167,171,175,179,181,185,187,189,195
%N A323703 Number of values of (X^3 + X) mod prime(n).
%C A323703 a(n) is also the number of values of any other polynomial of degree 3, except X^3.
%C A323703 a(n) appears to approach (2/3)*prime(n) as n increases.
%D A323703 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.
%H A323703 Alois P. Heinz, <a href="/A323703/b323703.txt">Table of n, a(n) for n = 1..10000</a>
%H A323703 Thomas Brazelton, Joshua Harrington, Matthew Litman, and Tony W. H. Wong, <a href="https://arxiv.org/abs/2103.09119">Distinct residues of Lucas polynomials over Fp</a>, arXiv:2103.09119 [math.NT], 2021. See p. 1.
%H A323703 Zhi-Hong Sun, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa84/aa8441.pdf">On the theory of cubic residues and nonresidues</a>, Acta Arithmetica 84.4 (1998): 291-335.
%H A323703 Zhi-Hong Sun, <a href="https://doi.org/10.1016/j.jnt.2005.10.012">On the number of incongruent residues of x^4 +ax^2 +bx modulo p</a>, Journal of Number Theory 119 (2006), 210-241. See p. 211.
%F A323703 a(n) = prime(n) - 2*floor(prime(n)/6 + 1/2), for n >= 3. - _Ridouane Oudra_, Jun 13 2020
%F A323703 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
%e A323703 a(1) = 1 since the only value X^3 + X takes mod 2 is 0.
%t A323703 Array[Length@ Union@ Mod[Array[#^3 + # &, #], #] &@ Prime@ # &, 62] (* _Michael De Vlieger_, Jan 27 2019 *)
%o A323703 (PARI) a(n) = #Set(vector(prime(n), k, Mod(k^3+k, prime(n)))); \\ _Michel Marcus_, Jan 25 2019
%Y A323703 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).
%K A323703 nonn
%O A323703 1,2
%A A323703 _Florian Severin_, Jan 24 2019