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.

7, 13, 31, 61, 67, 79, 97, 109, 139, 151, 157, 181, 193, 199, 211, 223, 241, 271, 277, 307, 349, 367, 373, 409, 433, 439, 547, 571, 601, 643, 661, 673, 733, 739, 751, 757, 769, 823, 907, 919, 937

Offset: 1

Author

Wolfdieter Lang, May 05 2016

Keywords

• nonn
• easy

Comments

The other part of this bisection appears in A272205.

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.

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, ...

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

eta^{12}(12*z) / (eta^4(6*z)*eta^4(24*z)) |A187076%20which%20gives%20the%20coefficients%20of%20the%20q-expansion%20of%20F(q)%20=%20Eta(q%5E%7B1/6%7D)%20/%20q%5E%7B1/6%7D%20=%20Product">{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.

Crossrefs

Cf. A001479, A001480, A002476, A047239, A187076, A272203, A272205 (complement relative to A002476).

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, 2 (mod 6), for m >= 1. A(m) = A001479(m+1).