

A272200


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 first case.


4



13, 19, 43, 61, 97, 103, 109, 127, 157, 163, 181, 193, 241, 277, 283, 331, 349, 373, 379, 409, 433, 463, 487, 499, 523, 601, 607, 619, 631, 661, 673, 691, 727, 733, 757, 769, 787, 811, 859, 883, 937, 967, 991
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OFFSET

1,1


COMMENTS

The other primes congruent to 1 modulo 3 are given in A272201.
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 such primes corresponding to A(m+1) == 1 (mod 3). The ones corresponding to A(m+1) not == 1 (mod 3) (the complement) are given in A272201.
This bisection of the primes from A002476 is needed in the formula for the coefficients of the qexpansion (q = exp(2*Pi*i*z), Im(z) > 0) of the modular cusp form (eta(6*z))^4_{z=z(q)} = Eta64(q) with Dedekind's eta function. See A000727 which gives the coefficients of the qexpansion of F(q) = Eta64(q^{1/6})/q^{1/6} = (Prod_{m>=0} (1  q^m))^4. The coefficients F(q) = Sum_{n >= 0} f(6*n+1)*q^n are given in the Finch link on p.5, using multiplicativity. For primes congruent to 1 modulo 6 the formula involves x_p and y_p which are the present A and B for prime p == 1 (mod 3).
See also the pdefects of the elliptic curve y^2 = x^3 + 1, related to (eta(6*z))^4, given in A272198 with another (equivalent) way to find the coefficients of the Eta64(q) expansion, hence those of F(q).


LINKS

Robert Israel, Table of n, a(n) for n = 1..10000
S. R. Finch, Powers of Euler's qSeries, arXiv:math/0701251 [math.NT], 2007.


FORMULA

This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) == 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] (* JeanFrançois Alcover, Apr 21 2020, after Robert Israel *)


CROSSREFS

Cf. A000727, A001479, A002476, A001480, A272198, A272201 (complement relative to A002476).
Sequence in context: A096455 A124199 A119869 * A106904 A106903 A098413
Adjacent sequences: A272197 A272198 A272199 * A272201 A272202 A272203


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Apr 28 2016


STATUS

approved



