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 A276664 Number of solutions to the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod p) as p runs through the primes. 3
 2, 1, 6, 9, 11, 11, 23, 15, 29, 23, 27, 35, 35, 33, 41, 59, 71, 59, 69, 59, 71, 87, 89, 95, 95, 95, 117, 101, 107, 119, 129, 131, 119, 135, 155, 171, 179, 153, 185, 179, 167, 191, 179, 167, 179, 207, 195, 213, 221, 215, 239, 215, 227, 251, 263, 245, 251, 291, 251 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176. FORMULA a(n) gives the number of solutions of the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod prime(n)), n >= 1. EXAMPLE The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2 == x^3 - x^2 + 4*x - 4 (mod prime(n)) begin: n, prime(n), a(n)\  solutions (x, y) 1,   2,       2:   (0, 0), (1, 0) 2,   3,       1:   (1, 0) 3,   5,       6:   (0, 1), (0, 4), (1, 0),                    (3, 1), (3, 4), (4, 0) 4,   7,       9:   (1, 0), (2, 1), (2, 6),                    (4, 2), (4, 5), (5, 2),                    (5, 5), (6, 2), (6, 5) PROG (Ruby) require 'prime' def A276664(n)   ary = []   Prime.take(n).each{|p|     a = Array.new(p, 0)     (0..p - 1).each{|i| a[(i * i) % p] += 1}     ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i - i * i + 4 * i - 4) % p]}   }   ary end CROSSREFS Sequence in context: A176013 A263255 A145663 * A160565 A025252 A177863 Adjacent sequences:  A276661 A276662 A276663 * A276665 A276666 A276667 KEYWORD nonn AUTHOR Seiichi Manyama, Sep 12 2016 STATUS approved

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Last modified June 25 21:43 EDT 2019. Contains 324357 sequences. (Running on oeis4.)