A235712 - OEIS

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.

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

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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: A161014 A344768 A363027 * A154852 A088996 A211888

Adjacent sequences: A235709 A235710 A235711 * A235713 A235714 A235715

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Apr 20 2014

STATUS

approved