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 A271228 #7 Sep 17 2016 11:53:07 %S A271228 0,0,0,-5,0,-7,0,-7,0,0,-11,-11,0,-13,0,0,0,13,5,0,10,4,0,0,-5,0,-7,0, %T A271228 -2,0,-19,0,0,7,0,-19,-25,-8,0,0,0,7,0,-23,0,28,13,-28,0,-22 %N A271228 P-defects p - N(p) of the elliptic curve y^2 = x^3 + 17 for primes p, where N(p) is the number of solutions modulo prime p. %C A271228 See A271227 for details and the conjecture for a(n) if prime(n) == 1 (mod 3). %C A271228 a(n) is negative for the 1 (mod 3) primes 7, 13, 19, 31, 37, 43, 97, 103, 109, 127, 151, 157, 163, 193, 223, 229, 241, 271, 277, 307, 313, 331, ... and positive for the primes 61, 67, 73, 79, 139, 181, 199, 211, 283, 337, 349, ... See A271227 for a comment on the conjectured three types I, II, and III of 1 (mod 3) primes. All three types appear for primes with negative as well as positive a(n) values. %D A271228 J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Table 45.5, Theorem 45.2, p. 400, Exercise 45.3, p. 404, p. 408 (4th ed., Pearson 2014, Table 5, Theorem 2, p. 366, Exercise 3, p. 370, p. 376) %H A271228 Seiichi Manyama, <a href="/A271228/b271228.txt">Table of n, a(n) for n = 1..10000</a> %F A271228 a(n) = prime(n) - A271227(n), where A271227(n) is the number of solutions of the congruence y^2 = x^3 + 17 (mod prime(n)). %F A271228 a(n) = 0 precisely for prime(n) == 0, or 2 (mod 3). See the Silverman reference, pp. 400 - 402 for the proof. (The case 0 (mod 3) is trivial.) %F A271228 Conjecture [WL]: For prime(n) = A002476(m) (a prime == 1 (mod 3)) one has a(n) = + or - sqrt(4*prime(n)) - 3*q(m)^2), with three alternative cases for q(m)^2, namely (2*B(m))^2, (A(m) - B(m))^2 and (A(m) + B(m))^2, where A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1. %e A271228 n = 4, prime(4) = 7, A271227(4) = 12 (see the example in A271227 for the solutions), a(4) = 7 - 12 = -5. Prime 7 is of type II. %e A271228 n = 25, prime(25) = 97, A271227(25) = 102, a(25) = -5. Prime 97 is of type III. %e A271228 n = 29, prime(29) = 109, A271227(29) = 111, a(29) = -2. Prime 109 is of type I. %e A271228 n = 18, prime(18) = 61, A271227(18) = 48, a(18) = +13. Prime 61 is of type II. %e A271228 n = 19, prime(19) = 67, A271227(19) = 62, a(19) = +5. Prime 67 is of type III. %e A271228 n = 21, prime(21) = 73, A271227(21) = 63, a(21) = +10. Prime 73 is of type I. %Y A271228 Cf. A271227. %K A271228 sign,easy %O A271228 1,4 %A A271228 _Wolfdieter Lang_, Apr 21 2016