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 A272196 #39 Sep 09 2024 09:35:45
%S A272196 2,4,4,9,10,9,19,19,24,29,24,34,49,49,39,59,54,49,74,74,69,89,89,74,
%T A272196 104,99,119,89,99,104,119,149,144,129,159,149,164,159,179,179,194,174,
%U A272196 174,189,199,199,199,204,209,214,209,269,249,274,259,249,259,299,279,299
%N A272196 Number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod p) as p runs through the primes.
%C A272196 This elliptic curve is discussed in the second Silverman reference. The present sequence is given in the rows named N_p in Table 45.6, p. 403. In the rows named a_p the p-defects prime(n) - a(n) are shown.
%C A272196 This sequence also gives the number of solutions of the congruences y^2 + y == x^3 - x^2 - 10*x - 20 (mod prime(n)) as well as y^2 + y == x^3 - x^2 (mod prime(n)) for n > 1 (cf. A376073). The first one is given in the Martin and Ono reference in Theorem 2, first row of the table, and the second one is given in the Frenkel reference, p. 84. (Of course, one could change the sign of y in both congruences.)
%C A272196 The modularity pattern for the elliptic curve y^2 = x^3 - 4*x^2 + 16 (and the ones mentioned in the previous comment and a comment below) is exhibited by the modular cusp form of weight 2 and level 11 (eta(z)*eta(11*z))^2, where eta is the Dedekind function, which in the q = exp(2*Pi*i*z), (Im(z) > 0) expansion has coefficients given in A006571 (with A006571(0) = 0). For all odd primes (2 is a bad prime), A002070(n) = A006571(prime(n)) = prime(n) - a(n), n >= 2, the p-defect. A006571(2) = -2, not 2-2 = 0. Note that the discriminant of this elliptic curve is -2^8*11 (sometimes -2^12*11 is used). Prime 11 is also bad for this curve, but A006571(11) = 1 = 11 - a(5) = 11 - 10. The curve y^2 + y = x^3 - x^2 - 10*x - 20 has discriminant -11^5 (see the first Silverman reference, pp. 46-48).
%C A272196 From _Wolfdieter Lang_, Jan 02 2017: (Start)
%C A272196 The congruence y^2 + y == x^3 - x^2 - 7820*x - 263580 (mod p) as p runs through the odd primes has the same number of solutions. See the Cremona link, N=11.
%C A272196 If b_n(Q) is the number of solutions of the Diophantine equation Q(x1,x2,x3,x4) = n with the quadratic form Q(x1,x2,x3,x4) = x1^2 + 4*(x2^2+x3^2+x4^2) + x1*x3 + 4*x2*x3 + 3*x2*x4 + 7*x3*x4 then the theta series delta(q;Q) = 1 + Sum_{n>=1} b_n(Q)*q^n equals (1/5)*E(q) + (18/5)*f(q) with the expansion coefficients of E(q) given by A185699 and those of f(q) = (eta(z)*eta(11*z))^2 with q = exp(2*Pi*i*z), (Im(z) > 0) given by A006571. See the Moreno-Wagstaff reference, pp. 245-246. b_n(Q), E(q) and f(q) are there denoted by a_n(Q), 12*E_{Chi0}(z) and f(z), respectively, and a missing n in the numerator of E_{Chi0}(z) has to be added (see A185699). (End)
%D A272196 Edward Frenkel, Liebe und Mathematik, Springer, Spektrum, 2014, p. 84.
%D A272196 Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York, pp. 246-247 (corrected).
%D A272196 J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 1986, pp. 46-48.
%D A272196 J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Table 45.6, p. 403, Theorem 47.2, p. 413 (4th ed., Pearson 2014, Table 6, p. 369, Theorem 2, p. 383)
%H A272196 Seiichi Manyama, <a href="/A272196/b272196.txt">Table of n, a(n) for n = 1..10000</a>
%H A272196 J. E. Cremona, <a href="https://homepages.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html">Algorithms for Modular Elliptic Curves</a>.
%H A272196 Yves Martin and Ken Ono, <a href="http://www.ams.org/journals/proc/1997-125-11/S0002-9939-97-03928-2/">Eta-Quotients and Elliptic Curves</a>, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
%H A272196 Carlos J. Moreno and Samuel S. Wagstaff, Jr., <a href="https://doi.org/10.1201/9781420057232">Sums of Squares of Integers</a>, Chapman & Hall/CRC, Boca Raton, London, New York, pp. 246-247.
%H A272196 J. H. Silverman, <a href="https://www.semanticscholar.org/paper/The-arithmetic-of-elliptic-curves-Silverman/7d62cc6267a4c9f513b45a874fdcd7d6582c0cdb">The Arithmetic of Elliptic Curves</a>, Springer, 1986, pp. 46-48.
%F A272196 a(n) gives the number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod prime(n)), n >= 1.
%e A272196 The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2 == x^3 - 4*x^2 + 16 (mod prime(n)) begin:
%e A272196 n, prime(n), a(n)\ solutions (x, y)
%e A272196 1, 2, 2: (0, 0), (1, 1)
%e A272196 2, 3, 4: (0, 1), (0, 2), (1, 1), (1, 2)
%e A272196 3, 5, 4: (0, 1), (0, 4), (4, 1), (4, 4)
%e A272196 4, 7, 9: (0, 3), (0, 4), (2, 1), (2, 6),
%e A272196 (4, 3), (4, 4), (6, 2), (6, 5)
%e A272196 5, 11, 10: (0, 4), (0, 7), (4, 4), (4, 7),
%e A272196 (6, 0), (7, 3), (7, 8), (9, 5),
%e A272196 (9, 6), (10, 0)
%e A272196 ...
%e A272196 --------------------------------------------------
%Y A272196 Cf. A002070, A006571, A060457, A185699, A376073.
%K A272196 nonn,easy
%O A272196 1,1
%A A272196 _Wolfdieter Lang_, Apr 22 2016