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.
%I A272207 #23 Apr 07 2020 21:48:51
%S A272207 2,5,6,5,11,11,23,23,17,23,35,35,35,53,53,59,47,59,65,83,71,71,77,95,
%T A272207 95,95,89,113,107,119,125,131,119,143,155,131,179,173,149,179,191,191,
%U A272207 203,167,179,191,227,233,233,215,239,263,227,251,263,281,251,251,251,275
%N A272207 Number of solutions to the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) as p runs through the primes.
%C A272207 In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the fourth row, starting with conductor 20, as a strong Weil curve for the weight 2 newform (eta(2*z)*eta(10*z))^2, symbolically 2^2 10^2, with Im(z) > 0, and the Dedekind eta function. See A030205 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z)*(eta(z)*eta(5*z))^2. For the q-expansion of (eta(2*z)*eta(10*z))^2 one has interspersed 0's: 0, 1, 0, -2, 0, -1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, -6, ... This modular cusp form of weight 2 appears as the 39th entry in Martin's Table I.
%C A272207 For the p-defect prime(n) - a(n) see A273163(n), n >= 1.
%C A272207 The discriminant of this elliptic curve is -400 = -2^4*5^2 (bad primes 2 and 5, also the prime divisors of the conductor).
%C A272207 The congruence y^2 == x^3 + x^2 - x has the same number of solutions modulo prime(n). See a comment on A030205. The discriminant equals +5.
%H A272207 Seiichi Manyama, <a href="/A272207/b272207.txt">Table of n, a(n) for n = 1..10000</a>
%H A272207 Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A272207 Yves Martin and Ken Ono, <a href="http://dx.doi.org/10.1090/S0002-9939-97-03928-2">Eta-Quotients and Elliptic Curves</a>, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
%F A272207 a(n) gives the number of solutions of the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)), n >= 1.
%F A272207 a(n) gives also the number of solutions of the congruence y^2 == x^3 + x^2 - x (mod prime(n)), n >= 1.
%e A272207 The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
%e A272207 The solutions (x, y) of y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)) begin:
%e A272207 n, prime(n), a(n)\ solutions (x, y)
%e A272207 1, 2, 2: (0, 0), (1, 0)
%e A272207 2, 3, 5: (0, 1), (0, 2), (1, 1),
%e A272207 (1, 2) (2, 0)
%e A272207 3, 5, 6: (0, 2), (0, 3), (1, 0),
%e A272207 (2, 2), (2, 3), (4, 0)
%e A272207 4, 7, 5: (0, 2), (0, 5), (4, 3),
%e A272207 (4, 4), (6, 0)
%e A272207 5, 11, 11: (0, 2), (0, 9), (4, 1),
%e A272207 (4, 10), (5, 3), (5, 8),
%e A272207 (6, 4), (6, 7), (9, 5),
%e A272207 (9, 6), (10, 0)
%e A272207 ...
%e A272207 The solutions (x, y) of y^2 == x^3 + x^2 - x (mod prime(n)) begin:
%e A272207 n, prime(n), a(n)\ solutions (x, y)
%e A272207 1, 2, 2: (0, 0), (1, 1)
%e A272207 2, 3, 5: (0, 0), (1, 1), (1, 2),
%e A272207 (2, 1) (2, 2)
%e A272207 3, 5, 6: (0, 0), (1, 1), (1, 4),
%e A272207 (2, 0), (4, 1), (4, 4)
%e A272207 4, 7, 5: (0, 0), (1, 1), (1, 6),
%e A272207 (6, 1), (6, 6)
%e A272207 5, 11, 11: (0, 0), (1, 1), (1, 10),
%e A272207 (3, 0), (6, 2), (6, 9),
%e A272207 (7, 0), (9, 3), (9, 8),
%e A272207 (10, 1), (10, 10)
%e A272207 ...
%Y A272207 Cf. A000040, A030205, A273163.
%K A272207 nonn,easy
%O A272207 1,1
%A A272207 _Wolfdieter Lang_, May 20 2016