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A177864
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a(n) is the smallest nontrivial quadratic residue modulo prime(n), for n >= 3.
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0
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4, 2, 3, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 3, 3, 4, 2, 2, 2, 3, 2, 2, 4, 2, 3, 3, 2, 2, 3, 2, 4, 4, 2, 3, 4, 2, 4, 3, 3, 2, 2, 4, 2, 4, 2, 3, 3, 2, 2, 2, 3, 2, 2, 4, 2, 3, 2, 4, 4, 4, 2, 2, 4, 4, 2, 3, 3, 2, 2, 2, 3, 4, 2, 4, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 4, 2, 3, 2, 2, 3, 4, 2, 4, 2, 4, 3
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OFFSET
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3,1
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COMMENTS
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There is no quadratic residue > 1 modulo the first or 2nd prime, so the sequence begins with a(3).
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LINKS
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FORMULA
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a(n) = 2 or 3 or 4 according as prime(n) == 1,7,9,15,17,23 or 11,13 or 3,5,19,21 (mod 24), respectively, for n > 2, by the quadratic reciprocity law and its supplements.
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EXAMPLE
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The quadratic residues modulo prime(3) = 5 are 1 and 4, so a(3) = 4.
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MATHEMATICA
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Flatten[Table[ Extract[Flatten[ Position[Table[JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]], {2}], {n, 3, 100}]]
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PROG
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(PARI) a(n, p=prime(n))=[2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 3, 0, 3, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2][p%24] \\ Charles R Greathouse IV, Jun 14 2022
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CROSSREFS
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Cf. A063987 (triangle in which the n-th row gives the quadratic residues modulo prime(n)), A053760 (smallest positive quadratic nonresidue modulo prime(n)).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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