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A235712 Least prime p < prime(n) with 2^p + 1 a quadratic nonresidue modulo prime(n), or 0 if such a prime p does not exist. 5
0, 2, 0, 2, 7, 2, 2, 5, 2, 11, 11, 2, 7, 2, 2, 2, 5, 5, 2, 5, 2, 5, 2, 5, 2, 7, 2, 2, 5, 2, 2, 13, 2, 5, 13, 5, 2, 2, 2, 2, 5, 11, 5, 2, 2, 7, 5, 2, 2, 23, 2, 7, 5, 5, 2, 2, 5, 5, 2, 7, 2, 2, 2, 5, 2, 2, 7, 2, 2, 5, 2, 7, 2, 2, 11, 2, 5, 2, 5, 5, 5, 7, 7, 2, 5, 2, 5, 2, 7, 2, 2, 7, 2, 13, 7, 2, 5, 5, 2, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: a(n) > 0 for all n > 3.

Note that 2^3 + 1 = 3^2 is a quadratic residue modulo any prime p > 3. Also, there is no prime p < prime(316) = 2089 with 2^p + 1 a primitive root modulo 2089.

See also A234972 and A235709 for similar conjectures.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

EXAMPLE

a(4) = 2 since 2^2 + 1 = 5 is a quadratic nonresidue modulo prime(4) = 7.

MATHEMATICA

Do[Do[If[JacobiSymbol[2^(Prime[k])+1, Prime[n]]==-1, Print[n, " ", Prime[k]]; Goto[aa]], {k, 1, n-1}];

Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 100}]

CROSSREFS

Cf. A000040, A098640, A234972, A235709.

Sequence in context: A111111 A185343 A161014 * A154852 A088996 A211888

Adjacent sequences:  A235709 A235710 A235711 * A235713 A235714 A235715

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Apr 20 2014

STATUS

approved

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Last modified May 13 12:16 EDT 2021. Contains 343839 sequences. (Running on oeis4.)