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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 (list; graph; refs; listen; history; text; internal format)
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} = (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 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

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

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Last modified January 20 20:35 EST 2020. Contains 331096 sequences. (Running on oeis4.)