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

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

Author

Carmine Suriano, Apr 22 2014

Keywords

• nonn
• frac

Examples

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

Crossrefs

Cf. A000040, A130290, A236959, A306591.

Programs

Maple

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

PARI

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

PARI

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

Python

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

Formula

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