A235709 Least prime p < prime(n) with 2^p - 1 a quadratic residue modulo prime(n), or 0 if such a number does not exist.

0, 0, 0, 0, 2, 2, 7, 3, 2, 3, 3, 2, 5, 5, 2, 3, 2, 2, 7, 2, 2, 5, 2, 23, 2, 5, 3, 2, 2, 3, 5, 2, 3, 3, 3, 5, 2, 11, 2, 5, 2, 2, 2, 2, 3, 3, 11, 3, 2, 2, 3, 2, 2, 2, 5, 2, 7, 3, 2, 3, 3, 5, 3, 2, 2, 3, 5, 2, 2, 2, 7, 2, 3, 2, 7, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 11, 5, 2, 2, 5, 2, 5, 2, 7, 5, 3, 2

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 4.

We have verified this for all n = 5, ..., 10^8.

Note that the conjecture in A234972 implies that for any prime p > 3 there is a prime q < p with 2^q - 1 a quadratic nonresidue modulo p.

Links

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

Example

a(8) = 3 since 2^3 - 1 = 7 is a quadratic residue modulo prime(8) = 19, but 2^2 - 1 = 3 is not.

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, A001348, A234972.

Sequence in context: A014840 A218756 A317329 * A341760 A342453 A370393

Adjacent sequences: A235706 A235707 A235708 * A235710 A235711 A235712

Keyword

nonn

Author

Zhi-Wei Sun, Apr 20 2014

Status

approved