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A272201 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the second case. 4
7, 31, 37, 67, 73, 79, 139, 151, 199, 211, 223, 229, 271, 307, 313, 337, 367, 397, 421, 439, 457, 541, 547, 571, 577, 613, 643, 709, 739, 751, 823, 829, 853, 877, 907, 919, 997 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The other primes congruent to 1 modulo 3 are given in A272200, where also more details are given.

Each prime == 1 (mod 3) has a unique representation A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1 (see also A001479). The present sequence gives these primes corresponding to A(m+1) not congruent 1 modulo 3. The ones corresponding to A(m+1) == 1 (mod 3) (the complement) are given in A272200.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

This sequence collects the 1 (mod 3) primes p(m) =  A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) not == 1 (mod 3), for m >= 1. A(m) = A001479(m+1).

MAPLE

filter:= proc(n) local S, x, y;

    if not isprime(n) then return false fi;

    S:= remove(hastype, [isolve(x^2+3*y^2=n)], negative);

    subs(S[1], x) mod 3 <> 1

end proc:

select(filter, [seq(i, i=7..1000, 6)]); # Robert Israel, Apr 29 2019

MATHEMATICA

filterQ[n_] := Module[{S, x, y}, If[!PrimeQ[n], Return[False]]; S = Solve[x > 0 && y > 0 && x^2 + 3 y^2 == n, Integers]; Mod[x /. S[[1]], 3] != 1];

Select[Range[7, 1000, 6], filterQ] (* Jean-Fran├žois Alcover, Apr 21 2020, after Robert Israel *)

CROSSREFS

Cf. A000727, A001479, A002476, A001480, A272198, A272200 (complement relative to A002476).

Sequence in context: A040064 A241101 A238664 * A325423 A309381 A276741

Adjacent sequences:  A272198 A272199 A272200 * A272202 A272203 A272204

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Apr 28 2016

STATUS

approved

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Last modified April 23 12:15 EDT 2021. Contains 343204 sequences. (Running on oeis4.)