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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A272205 A bisection of the primes congruent to 1 modulo 3 (A002476). This is the part depending on the corresponding A001479 entry being congruent to 4 or 5 modulo 6.

Original entry on oeis.org

19, 37, 43, 73, 103, 127, 163, 229, 283, 313, 331, 337, 379, 397, 421, 457, 463, 487, 499, 523, 541, 577, 607, 613, 619, 631, 691, 709, 727, 787, 811, 829, 853, 859, 877, 883, 967, 991, 997
Offset: 1

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Author

Wolfdieter Lang, May 05 2016

Keywords

Comments

The other part of this bisection appears in 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 such primes corresponding to A(m) == 4, 5 (mod 6). The ones corresponding to A(m) == 1, 2 (mod 6) (the complement) are given in A272205.
The corresponding A001479 entries are 4, 5, 4, 5, 10, 10, 4, 11, 16, 11, 16, 17, 4, 17, 11, 5, 10, 22, 16, 4, 23, 23, 10, 5, 16, 22, 4, 11, 22, 28, 28, 23, 29, 28, 17, 4, 10, 22, 5, ...
See A272204 for a comment on the relevance of this bisection in connection with the signs of the q-expansion coefficients of the modular cusp form eta^{12}(12*z) / (eta^4(6*z)*eta^4(24*z)).

Crossrefs

Cf. A001479, A001480, A002476, A047239, A187076, A272203, A272204 (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) == 4, 5 (mod 6), for m >= 1. A(m) = A001479(m+1).

A272202 Number of solutions of the congruence y^2 == x^3 - 1 (mod p) as p runs through the primes.

Original entry on oeis.org

2, 3, 5, 3, 11, 11, 17, 27, 23, 29, 27, 47, 41, 51, 47, 53, 59, 47, 51, 71, 83, 75, 83, 89, 83, 101, 123, 107, 107, 113, 147, 131, 137, 123, 149, 147, 143, 171, 167, 173, 179, 155, 191, 191, 197, 171, 195, 195, 227, 251, 233, 239, 227, 251, 257, 263, 269, 243, 251, 281
Offset: 1

Views

Author

Wolfdieter Lang, May 05 2016

Keywords

Comments

In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the last column, starting with Conductor 144, as a strong Weil curve for the weight 2 newform eta^{12}(12*z) / (eta^4(6*z) * eta^4(24*z)), symbolically 12^{12} 6^{-4} 24^{-4}, with Im(z) > 0, and the Dedekind eta function. See A187076 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z/3)* eta(2*z)^{12} / (eta^4(z)*eta^4(4*z)). For the q-expansion of 12^{12} 6^{-4} 24^{-4} one has a leading zero and 5 interspersed 0's: 0,1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,-8,...
The discriminant of this elliptic curve is -3^3 = -27.
For the elliptic curve y^2 == x^3 + 1 (mod prime(n)) see A000727, A272197, A272198, A272200 and A272201.

Examples

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2 == x^3 - 1 (mod prime(n)) begin:
n, prime(n), a(n)\ solutions (x, y)
1,    2,      2:   (0, 1), (1, 0)
2,    3,      3:   (1, 0), (2, 1), (2, 2)
3,    5,      5:   (0, 2), (0, 3), (1, 0),
                   (3, 1), (3, 4)
4,    7,      3:   (1, 0), (2, 0), (4, 0)
5,    11,    11:   (1, 0), (3, 2), (3, 9),
                   (5, 5), (5, 6), (7, 1),
                   (7, 10), (8, 4), (8, 7),
                   (10, 3), (10, 8)
...

Links

Crossrefs

Cf. A000040, A187076, A272203.

Formula

a(n) gives the number of solutions of the congruence y^2 == x^3 - 1 (mod prime(n)), n >= 1.

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.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, May 05 2016

Keywords

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).
Showing 1-3 of 3 results.

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