cp's OEIS Frontend

A272204 A bisection of the primes congruent to 1 modulo 3 (A002476). This is the part depending on the corresponding A001479 entry being congruent to 1 or 2 modulo 6.

%I A272204 #13 Nov 26 2016 03:01:02
%S A272204 7,13,31,61,67,79,97,109,139,151,157,181,193,199,211,223,241,271,277,
%T A272204 307,349,367,373,409,433,439,547,571,601,643,661,673,733,739,751,757,
%U A272204 769,823,907,919,937
%N A272204 A bisection of the primes congruent to 1 modulo 3 (A002476). This is the part depending on the corresponding A001479 entry being congruent to 1 or 2 modulo 6.
%C A272204 The other part of this bisection appears in A272205.
%C A272204 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. The present sequence gives all such primes corresponding to A(m) == 1, 2 (mod 6). The ones corresponding to A(m) not == 1, 2 (mod 6) (the complement), that is == 4, 5 (mod 6), are given in A272205.
%C A272204 The corresponding A001479 entries are 2, 1, 2, 7, 8, 2, 7, 1, 8, 2, 7, 13, 1, 14, 8, 14, 7, 14, 13, 8, 7, 2, 19, 19, 1, 14, 20, 8, 13, 20, 19, 25, 25, 8, 26, 13, 1, 26, 20, 26, 13, ...
%C A272204 This bisection of the 1 (mod 3) primes A002476 is needed to determine the sign in the formula for the coefficients of the q-expansion (q = exp(2*Pi*i*z), Im(z) > 0) of the modular weight 2 cusp form
%C A272204   eta^{12}(12*z) / (eta^4(6*z)*eta^4(24*z)) |_{z=z(q)} =: Eta(q) with Dedekind's eta function. See A187076 which gives the coefficients of the q-expansion of F(q) = Eta(q^{1/6}) / q^{1/6} = Product_{m>=0} (1 - q^(2*m))^{12} / ((1 - q^m)*(1 - q^(4*m)))^4. The q-expansion coefficients (called b(n)) of the modular cusp form are given there using multiplicativity. Note that there x can also be negative, whereas here A is positive.
%F A272204 This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) == 1, 2 (mod 6), for m >= 1. A(m) = A001479(m+1).
%Y A272204 Cf. A001479, A001480, A002476, A047239, A187076, A272203, A272205 (complement relative to A002476).
%K A272204 nonn,easy
%O A272204 1,1
%A A272204 _Wolfdieter Lang_, May 05 2016