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A330423 Least nonsquare k that is a quadratic residue modulo n and is coprime to n.

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%I A330423 #10 Dec 18 2019 05:29:27
%S A330423 2,3,7,5,6,7,2,17,7,11,3,13,3,11,19,17,2,7,5,21,22,3,2,73,6,3,7,29,5,
%T A330423 19,2,17,31,13,11,13,3,5,10,41,2,37,6,5,19,3,2,73,2,11,13,17,6,7,14,
%U A330423 57,7,5,3,61,3,5,22,17,14,31,6,13,13,11,2,73,2,3,19,5,15
%N A330423 Least nonsquare k that is a quadratic residue modulo n and is coprime to n.
%C A330423 a(n) > n if and only if n is in A303704.
%C A330423 It seems that lim_{n->oo} a(n)/n = 0. Conjectured last term m such that a(m)/m > 1/k, k = 1, 2, 3, ...: 840, 1680, 2640, 9240, 10920, 10920, 18480, 18480, 21840, 29640, ...
%H A330423 Jianing Song, <a href="/A330423/b330423.txt">Table of n, a(n) for n = 1..10000</a>
%e A330423 k is a coprime quadratic residue modulo 16 if and only if k == 1 (mod 8). Since 1 and 9 are squares, a(16) = 17.
%e A330423 k is a coprime quadratic residue modulo 24 if and only if k == 1 (mod 24). Since 1, 25 and 49 are squares, a(24) = 73.
%e A330423 k is a coprime quadratic residue modulo 840 if and only if k == 1, 121, 169, 289, 361, 529 (mod 840). Since 1, 121, 169, 289, 361, 529, 841, 961 are squares, a(840) = 840+169 = 1009.
%o A330423 (PARI) a(n) = my(k=1); while(!issquare(Mod(k,n)) || issquare(k) || gcd(k,n)>1, k++); k
%Y A330423 Cf. A303704, A330404.
%K A330423 nonn
%O A330423 1,1
%A A330423 _Jianing Song_, Dec 14 2019