A236959 -id:A236959 - cp's OEIS frontend

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

A236965 Number of nonzero quartic residues modulo the n-th prime.

Original entry on oeis.org

1, 1, 1, 3, 5, 3, 4, 9, 11, 7, 15, 9, 10, 21, 23, 13, 29, 15, 33, 35, 18, 39, 41, 22, 24, 25, 51, 53, 27, 28, 63, 65, 34, 69, 37, 75, 39, 81, 83, 43, 89, 45, 95, 48, 49, 99, 105, 111, 113, 57, 58, 119, 60, 125, 64, 131, 67, 135, 69, 70, 141, 73, 153, 155, 78

Offset: 1

Carmine Suriano, Apr 22 2014

			a(5) = 5 for x^4 (mod 11 = prime(5)) equals 1, 3, 4, 5, 9.

Cf. A000040, A130290, A236959, A306591.

seq((ithprime(n)-1)/gcd(ithprime(n)-1,4), n=1..80); # Ridouane Oudra, Mar 13 2025

a(n) = numerator(1/2 - 1/(prime(n)+1)); \\ Michel Marcus, Feb 26 2019

a(n) = my(p=prime(n)); sum(k=0, p-1, m = Mod(k,p); m && ispower(Mod(k,p), 4)); \\ Michel Marcus, Feb 26 2019

from sympy import prime
from fractions import Fraction
def a(n): return (Fraction(1, 2) - Fraction(1, (prime(n)+1))).numerator
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Jun 04 2021

For odd primes, if prime(n) = 4k+1 then a(n) = (prime(n)-1)/4, if prime(n) = 4k+3 then a(n)=(prime(n)-1)/2.
a(n) = numerator(1/2 - 1/(prime(n)+1)). - Michel Marcus, Feb 26 2019
a(n) = A006093(n)/gcd(A006093(n),4). - Ridouane Oudra, Mar 13 2025

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

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A236965 Number of nonzero quartic residues modulo the n-th prime.

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