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In Combinatorial Game Theory, the nim product of two ordinals is defined by:

a*b = the least ordinal not equal to any a*b' + a'*b + a'*b' with a' < a, b' < b.

Here + is nim addition (binary xor). With this definition, the ordinals form a Field ON_2 of characteristic 2.

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.

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.

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.

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.

a(1)-a(3): John H. Conway

a(4)-a(13): Hendrik W. Lenstra

a(14)-a(18): Lieven Le Bruyn

a(19)-a(59): Aaron N. Siegel

A partizan combinatorial game G is a left dead end if no subposition of G has any left options. In normal play, every left dead end is equal to a nonpositive integer. In misère play, the left dead ends have a more intricate structure; this sequence counts the misère-inequivalent left dead ends with birthday <= n.

As A354382, but with mirror-image figures counted as distinct.

As A354380, but with mirror-image figures counted as distinct.

The impartial combinatorial game Mem0 (aka Short Local Nim) is played with heaps of tokens, as in Nim. On each turn, k tokens may be removed from a heap H, provided that k is not equal to the number of tokens that were removed on the immediately preceding move on H.

A heap may be denoted by n_k, where n is the number of tokens remaining and k the number removed on the preceding move. There are many nim values m that occur at just finitely many heap sizes, in the sense that G(n_k) = m for just finitely many choices of n. This sequence gives the exceptional values of m that occur at infinitely many heap sizes.

It is unknown whether there are infinitely many such m. It is remarkable that such simple, parameterless rules give rise to an unusual and mysterious integer sequence.

See A057787 for a description of polyarcs. The pseudo-polyarcs are constructed in the same way as ordinary polyarcs, but allowing for corner-connections. Thus they generalize polyarcs in the same way that pseudo-polyominoes (aka polyplets, A030222) generalize ordinary polyominoes (A000105). They can also be viewed as the "rounded" variant of pseudo-polytans (A354380), in the same way that ordinary polyarcs are the rounded variant of ordinary polytans (A006074).

Two figures are considered equivalent if they differ only by a rotation or reflection.

The pseudo-polyarcs grow tremendously fast, much faster than most polyforms. The initial data that have been computed suggest an asymptotic growth rate of at least 36^n.

A pseudo-polytan is a planar figure consisting of n isosceles right triangles joined either edge-to-edge or corner-to-corner, in such a way that the short edges of the triangles coincide with edges of the square lattice. Two figures are considered equivalent if they differ only by a rotation or reflection.

The pseudo-polytans are constructed in the same way as ordinary polytans (A006074), but allowing for corner-connections. Thus they generalize polytans in the same way that pseudo-polyominoes (aka polyplets, A030222) generalize ordinary polyominoes (A000105).