%I #21 Oct 24 2018 08:07:57
%S 2,1,6,9,11,11,23,15,29,23,27,35,35,33,41,59,71,59,69,59,71,87,89,95,
%T 95,95,117,101,107,119,129,131,119,135,155,171,179,153,185,179,167,
%U 191,179,167,179,207,195,213,221,215,239,215,227,251,263,245,251,291,251
%N Number of solutions to the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod p) as p runs through the primes.
%C This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, (eta(4t)*eta(20t))^6 / (eta(2t)*eta(8t)*eta(10t)*eta(40t))^2, see Theorem 2 in Martin & Ono.
%H Seiichi Manyama, <a href="/A276664/b276664.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 a(n) gives the number of solutions of the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod prime(n)), n >= 1.
%e The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
%e The solutions (x, y) of y^2 == x^3 - x^2 + 4*x - 4 (mod prime(n)) begin:
%e n, prime(n), a(n)\ solutions (x, y)
%e 1, 2, 2: (0, 0), (1, 0)
%e 2, 3, 1: (1, 0)
%e 3, 5, 6: (0, 1), (0, 4), (1, 0),
%e (3, 1), (3, 4), (4, 0)
%e 4, 7, 9: (1, 0), (2, 1), (2, 6),
%e (4, 2), (4, 5), (5, 2),
%e (5, 5), (6, 2), (6, 5)
%o (Ruby)
%o require 'prime'
%o def A276664(n)
%o ary = []
%o Prime.take(n).each{|p|
%o a = Array.new(p, 0)
%o (0..p - 1).each{|i| a[(i * i) % p] += 1}
%o ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i - i * i + 4 * i - 4) % p]}
%o }
%o ary
%o end
%K nonn
%O 1,1
%A _Seiichi Manyama_, Sep 12 2016
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