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A057128 Numbers n such that -3 is a square mod n.

%I A057128 #21 Mar 05 2020 17:00:32
%S A057128 1,2,3,4,6,7,12,13,14,19,21,26,28,31,37,38,39,42,43,49,52,57,61,62,67,
%T A057128 73,74,76,78,79,84,86,91,93,97,98,103,109,111,114,122,124,127,129,133,
%U A057128 134,139,146,147,148,151,156,157,158,163,169,172,181,182,183,186,193
%N A057128 Numbers n such that -3 is a square mod n.
%C A057128 The fact that there are no numbers in this sequence of the form 6k+5 leads to the result that all prime factors of central polygonal numbers (A002061 of the form n^2-n+1) are either 3 or of the form 6k+1. This in turn leads to there being an infinite number of primes of the form 6k+1, since if P=product[all known primes of form 6k+1] then all the prime factors of 9P^2-3P+1 must be unknown primes of form 6k+1.
%C A057128 Numbers that are not multiples of 8 or 9 and for which all prime factors greater than 3 are congruent to 1 mod 6. - _Eric M. Schmidt_, Apr 21 2013
%C A057128 Numbers that divide at least some member of A117950. - _Robert Israel_, Feb 19 2016
%H A057128 Eric M. Schmidt, <a href="/A057128/b057128.txt">Table of n, a(n) for n = 1..1000</a>
%e A057128 a(7)=13 since -3 mod 13=10 mod 13=6^2 mod 13.
%p A057128 select(t -> numtheory:-quadres(-3,t) = 1, {$1..1000}); # _Robert Israel_, Feb 19 2016
%t A057128 Select[Range[200], IntegerQ[PowerMod[-3, 1/2, #]]&] // Quiet (* _Jean-François Alcover_, Mar 05 2019 *)
%o A057128 (Sage)
%o A057128 def A057128(n) :
%o A057128     if n%8==0 or n%9==0: return False
%o A057128     for (p, m) in factor(n) :
%o A057128         if p % 6 not in [1, 2, 3] : return False
%o A057128         return True
%o A057128 # _Eric M. Schmidt_, Apr 21 2013
%o A057128 (PARI) isok(n) = issquare(Mod(-3,n)); \\ _Michel Marcus_, Feb 19 2016
%Y A057128 Includes the primes in A045331 and these (primes congruent to {1, 2, 3} mod 6) are the prime factors of the terms in this sequence. Cf. A008784, A057125, A057126, A057127, A057129.
%Y A057128 Cf. A117950.
%K A057128 nonn
%O A057128 1,2
%A A057128 _Henry Bottomley_, Aug 10 2000