A057128 - OEIS

%I #21 Mar 05 2020 17:00:32

%S 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 73,74,76,78,79,84,86,91,93,97,98,103,109,111,114,122,124,127,129,133,

%U 134,139,146,147,148,151,156,157,158,163,169,172,181,182,183,186,193

%N Numbers n such that -3 is a square mod n.

%C 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 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 Numbers that divide at least some member of A117950. - _Robert Israel_, Feb 19 2016

%H Eric M. Schmidt, <a href="/A057128/b057128.txt">Table of n, a(n) for n = 1..1000</a>

%e a(7)=13 since -3 mod 13=10 mod 13=6^2 mod 13.

%p select(t -> numtheory:-quadres(-3,t) = 1, {$1..1000}); # _Robert Israel_, Feb 19 2016

%t Select[Range[200], IntegerQ[PowerMod[-3, 1/2, #]]&] // Quiet (* _Jean-François Alcover_, Mar 05 2019 *)

%o (Sage)

%o def A057128(n) :

%o     if n%8==0 or n%9==0: return False

%o     for (p, m) in factor(n) :

%o         if p % 6 not in [1, 2, 3] : return False

%o         return True

%o # _Eric M. Schmidt_, Apr 21 2013

%o (PARI) isok(n) = issquare(Mod(-3,n)); \\ _Michel Marcus_, Feb 19 2016

%Y 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 Cf. A117950.

%K nonn

%O 1,2

%A _Henry Bottomley_, Aug 10 2000