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A222717 revision #2

A222717

Primes p whose smallest positive quadratic nonresidue is not a primitive root of p.

41, 43, 103, 109, 151, 157, 191, 229, 251, 271, 277, 283, 307, 311, 313, 331, 337, 367, 397, 409, 439, 457, 499, 571, 643, 683, 691, 727, 733, 739, 761, 769, 811, 911, 919, 967, 971, 991, 997, 1013, 1021, 1031, 1051, 1069, 1093, 1151, 1163, 1181, 1289, 1297, 1303, 1321, 1399, 1429, 1459, 1471, 1489, 1543, 1559, 1579, 1597, 1613, 1627, 1657, 1699, 1709, 1723, 1753, 1759, 1783, 1789, 1811, 1871, 1873, 1879, 1933

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OFFSET

1,1

COMMENTS

Same as primes p such that if q is the smallest positive quadratic nonresidue mod p, then q^k == 1 mod p for some positive integer k < p-1.

A primitive root of a prime p is always a quadratic nonresidue mod p. (Proof. If g == x^2 mod p, then g^((p-1)/2) == x^(p-1) == 1 mod p, and so g is not a primitive root of p.) But a quadratic nonresidue mod p may or may not be a primitive root of p.

See A001918 (least positive primitive root of the n-th prime) and A053760 (smallest positive quadratic nonresidue of the n-th prime) for references and additional comments and links.

LINKS

Table of n, a(n) for n=1..76.

Index entries for primes by primitive root

EXAMPLE

The smallest positive quadratic nonresidue of 41 is 3, and 3 is not a primitive root of 41, so 41 is a member.

MATHEMATICA

NR = (Table[p = Prime[n]; First[ Select[ Range[p], JacobiSymbol[#, p] != 1 &]], {n, 1, 300}]); Select[ Prime[ Range[300]], MultiplicativeOrder[ NR[[ PrimePi[#]]], #] < # - 1 &]

CROSSREFS

Cf. A001918, A053760.

Sequence in context: A180548 A157194 A161614 * A106098 A155500 A185657

Adjacent sequences: A222714 A222715 A222716 * A222718 A222719 A222720

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Mar 12 2013

STATUS

editing