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 A380496 #15 Jan 26 2025 20:53:58
%S A380496 0,0,1,1,0,0,4,1,0,1,0,1,1,1,1,1,0,0,0,1,1,1,1,0,1,0,1,0,0,1,0,1,0,1,
%T A380496 0,1,4,1,0,1,0,0,0,0,0,0,1,1,1,1,0,0,1,0,1,1,0,1,0
%N A380496 Lenstra excess of the n-th odd prime.
%C A380496 In Combinatorial Game Theory, the nim product of two ordinals is defined by:
%C A380496 a*b = the least ordinal not equal to any a*b' + a'*b + a'*b' with a' < a, b' < b.
%C A380496 Here + is nim addition (binary xor). With this definition, the ordinals form a Field ON_2 of characteristic 2.
%C A380496 Conway showed that under nim addition and nim multiplication, the ordinals below w^w^w form an algebraic and algebraically closed subfield of ON_2, i.e., they form the algebraic closure of {0,1}. (Here w = omega = the least infinite ordinal.) Conway moreover gave a description of the arithmetic of ordinals below w^w^w. This arithmetic depends on calculating a particular ordinal alpha_p for each odd prime p: specifically, if p is the (n+1)-st odd prime, then alpha_p is defined to be the p-th nim-power of w^w^n. It is always the case that alpha_p < w^w^n.
%C A380496 Lenstra later showed that for each such p, there is a particular ordinal kappa_{f(p)} (following Lenstra's notation) such that alpha_p = kappa_{f(p)} + m_p for some integer m_p >= 0. This integer m_p is the Lenstra excess of p. It is usually 0 or 1, with the only other observed values for p <= 281 being m_19 = m_163 = 4.
%C A380496 Lenstra gave an algorithm for calculating m_p, but the values are in general quite hard to compute. The calculation depends on carrying out operations in the finite subfield F_p of ON_2 generated by w^w^n. The size of F_p is always 2^(e_p) for some integer e_p (the Lenstra exponent of p). The running time of Lenstra's algorithm is on the order of O(e_p^3), and the values of e_p, while erratic, tend to grow exponentially in p. For p <= 281 the largest exponent is e_263 = 102180; whereas for p = 283 (the least prime for which m_p is unknown as of January 2025), we have e_283 = 237820.
%C A380496 The latest version of CGSuite implements the arithmetic of w^w^w and includes Scala code for calculating the values of m_p and alpha_p.
%C A380496 a(1)-a(3): John H. Conway
%C A380496 a(4)-a(13): Hendrik W. Lenstra
%C A380496 a(14)-a(18): Lieven Le Bruyn
%C A380496 a(19)-a(59): _Aaron N. Siegel_
%D A380496 John H. Conway, On Numbers and Games, second edition. A K Peters, Ltd. / CRC Press, Natick, MA, 2001.
%D A380496 Hendrik W. Lenstra, On the algebraic closure of two, Proc. Kon. Ned. Akad. Wet. Series A 80 (1977), 389-396
%D A380496 Aaron N. Siegel, Combinatorial Game Theory. Number 146 in Graduate Studies in Mathematics. American Mathematical Society, 2013.
%H A380496 <a href="https://www.cgsuite.org">Combinatorial Game Suite</a>
%H A380496 neverendingbooks, <a href="http://www.neverendingbooks.org/on2-extending-lenstras-list">On2, Extending Lenstra's List</a>, January 27 2009.
%e A380496 For n <= 4 the corresponding ordinals alpha_p are:
%e A380496 alpha_3 = 2,
%e A380496 alpha_5 = 4,
%e A380496 alpha_7 = w + 1,
%e A380496 alpha_11 = w^w + 1.
%K A380496 nonn,hard,more
%O A380496 1,7
%A A380496 _Aaron N. Siegel_, Jan 21 2025