Florian Severin - cp's OEIS frontend

User: Florian Severin

Florian Severin has authored 2 sequences.

A323704 Number of cubic residues (including 0) modulo the n-th prime.

2, 3, 5, 3, 11, 5, 17, 7, 23, 29, 11, 13, 41, 15, 47, 53, 59, 21, 23, 71, 25, 27, 83, 89, 33, 101, 35, 107, 37, 113, 43, 131, 137, 47, 149, 51, 53, 55, 167, 173, 179, 61, 191, 65, 197, 67, 71, 75, 227, 77, 233, 239, 81, 251, 257, 263, 269, 91, 93, 281, 95, 293

Offset: 1

Florian Severin, Jan 24 2019

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046530, A000040, A236959.

Table[With[{p=Prime[n]},If[Mod[p,3]==1,1+(p-1)/3,p]],{n,80}] (* Harvey P. Dale, Feb 02 2025 *)

a(n) = my(p=prime(n)); sum(k=0, p-1, ispower(Mod(k,p), 3)); \\ Michel Marcus, Feb 26 2019

from sympy import prime
def a(n):
  p = prime(n)
  return len(set([x**3 % p for x in range(p)]))

If prime(n) - 1 = 3k then a(n) = k+1, otherwise a(n) = prime(n). (Cf. formula for A236959.)
a(n) = A236959(n) + 1.
a(n) = A046530(A000040(n)). - Rémy Sigrist, Jan 24 2019

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

Original entry on oeis.org

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

Florian Severin, Jan 24 2019

nonn

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.

			a(1) = 1 since the only value X^3 + X takes mod 2 is 0.

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.

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.

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).

Array[Length@ Union@ Mod[Array[#^3 + # &, #], #] &@ Prime@ # &, 62] (* Michael De Vlieger, Jan 27 2019 *)

a(n) = #Set(vector(prime(n), k, Mod(k^3+k, prime(n)))); \\ Michel Marcus, Jan 25 2019

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

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User: Florian Severin

A323704 Number of cubic residues (including 0) modulo the n-th prime.

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