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.

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

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 q-expansion (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 q-expansion of F(q) = Eta64(q^(1/6))/q^(1/6) = (Product_{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 p-defects 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 q-Series, 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] (* Jean-François 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