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 A057128 Numbers n such that -3 is a square mod n. 8
 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, 73, 74, 76, 78, 79, 84, 86, 91, 93, 97, 98, 103, 109, 111, 114, 122, 124, 127, 129, 133, 134, 139, 146, 147, 148, 151, 156, 157, 158, 163, 169, 172, 181, 182, 183, 186, 193 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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 Numbers that divide at least some member of A117950. - Robert Israel, Feb 19 2016 LINKS Eric M. Schmidt, Table of n, a(n) for n = 1..1000 EXAMPLE a(7)=13 since -3 mod 13=10 mod 13=6^2 mod 13. MAPLE select(t -> numtheory:-quadres(-3, t) = 1, {\$1..1000}); # Robert Israel, Feb 19 2016 MATHEMATICA Select[Range[200], IntegerQ[PowerMod[-3, 1/2, #]]&] // Quiet (* Jean-François Alcover, Mar 05 2019 *) PROG (Sage) def A057128(n) : if n%8==0 or n%9==0: return False for (p, m) in factor(n) : if p % 6 not in [1, 2, 3] : return False return True # Eric M. Schmidt, Apr 21 2013 (PARI) isok(n) = issquare(Mod(-3, n)); \\ Michel Marcus, Feb 19 2016 CROSSREFS 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. Cf. A117950. Sequence in context: A239014 A030705 A305929 * A018534 A018276 A057732 Adjacent sequences: A057125 A057126 A057127 * A057129 A057130 A057131 KEYWORD nonn AUTHOR Henry Bottomley, Aug 10 2000 STATUS approved

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Last modified December 2 09:50 EST 2023. Contains 367517 sequences. (Running on oeis4.)