A243901 Positive integers n such that p_{n+i} is a quadratic residue modulo p_{n+j} for any distinct i and j among 0, 1, ..., 6

178633, 2355662, 4892172, 5702347, 9256159, 9572343, 13837265, 15147032, 15429648, 15822376, 16603935, 20925043, 22128672, 22462201, 22689295, 27145167, 28031877, 28470899, 29246422, 30772941, 31211796, 32372758

Offset

1,1

Comments

Conjecture: For any integer m > 0, there are infinitely many positive integers n such that p_{n+i} is a quadratic residue modulo p_{n+j} for any distinct i and j among 0, 1, ..., m.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..200

Hao Pan, Z.-W. Sun, Consecutive primes and Legendre symbols, arXiv preprint arXiv:1406.5951 [math.NT], 2014-2018.

Example

a(1) = 178633 since any 6 primes among the 7 integers prime(178633) = 2434589, prime(178634) = 2434609, prime(178635) = 2434613, prime(178636) = 2434657, prime(178637) = 2434669, prime(178638) = 2434673 and prime(178639) = 2434681 are quadratic residues modulo the remaining one of the 7 primes.

Mathematica

q[i_, j_]:=JacobiSymbol[Prime[i], Prime[j]]
m=0; Do[Do[If[q[n+i, n+j]==-1, Goto[aa]], {i, 0, 6}, {j, 0, 6}]; m=m+1; Print[m, " ", n]; Label[aa]; Continue, {n, 1, 32372758}]
Reap[ Do[ If[ Catch[ Do[ If[ JacobiSymbol[Prime[n + i], Prime[n + j]] != 1, Throw@False], {i, 0, 5}, {j, i + 1, 6}]; True], Sow[n]], {n, 32372758}]][[2, 1]] (* Michael Somos, Jun 15 2014 *)

Crossrefs

Cf. A000040, A243755, A243837, A243839.

Sequence in context: A133972 A233480 A233475 * A254801 A254808 A253817

Adjacent sequences: A243898 A243899 A243900 * A243902 A243903 A243904

Keyword

nonn

Author

Zhi-Wei Sun, Jun 14 2014

Status

approved