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%I #25 Aug 05 2019 02:01:23
%S 178633,2355662,4892172,5702347,9256159,9572343,13837265,15147032,
%T 15429648,15822376,16603935,20925043,22128672,22462201,22689295,
%U 27145167,28031877,28470899,29246422,30772941,31211796,32372758
%N 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
%C 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.
%H Zhi-Wei Sun, <a href="/A243901/b243901.txt">Table of n, a(n) for n = 1..200</a>
%H Hao Pan, Z.-W. Sun, <a href="http://arxiv.org/abs/1406.5951">Consecutive primes and Legendre symbols</a>, arXiv preprint arXiv:1406.5951 [math.NT], 2014-2018.
%e 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.
%t q[i_,j_]:=JacobiSymbol[Prime[i],Prime[j]]
%t 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}]
%t 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 *)
%Y Cf. A000040, A243755, A243837, A243839.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jun 14 2014
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