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A236965
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Number of nonzero quartic residues modulo the n-th prime.
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1
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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
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OFFSET
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1,4
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LINKS
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FORMULA
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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
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EXAMPLE
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a(5)=5 for x^4 (mod 11=prime(5)) equals 1, 3, 4, 5, 9.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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