cp's OEIS Frontend

For the rules of this two person game with cards labeled from 1 to n, for n >= 1, called JG(n), see the Ian Stewart links.

It is reported (see the FEEDBACK and the German version), that E. P. Wigner used this game in some lecture in the thirties. There the prime factorization of n! into prime powers, with the number of odd or even (>= 2) exponents, seems to have played a role (see A055460(n) and A348841(n) for the number of primes with these exponents in the factorization of n!, respectively).

The repertoire of card numbers for JG(n) that can be chosen if the latest removed card had label k is shown in A348390. Of course, only those card numbers not yet removed in earlier moves qualify. E.g., n = 4, k = 2: repertoire 1, 4.

The total number of games JG(n), for n >= 2, if the first removed card has label K = 2*k, for k = 1, 2, ... ,floor(n/2), is given in A348843.

For the irregular table which gives in row n the odd and even number of moves in the a(n) JG(n) games see A348844. This gives the number of times Alice (the first mover), respectively Bob wins.

The Ian Stewart links for the Juniper Green game are given in A348842.

The length of row n is A008619(n-2), for n >= 2.

The sum of row n gives A348842(n).

See A348844 for the table where the present entries are split into the number of games with odd and even number of moves.

T(n, K=2*k) is determined from the sequence of removed numbers in the Julian Green game with cards labeled 1, 2, ..., n, named JG(n), if the first removed card number is K. Such a sequence is obtained from row n of table A348390 which provides the possible card numbers (the reservoir) for the move following the sequence entry with card number q, for q from {1, 2, ..., n} (q is here the number k of A348390, giving the sequence of proper divisors of d(n,k) followed by the sequence of multiples m(n, k) of k which are > k and <= n). However, numbers of the reservoir only qualify if they have not yet been removed in the game. E.g., a JG(5) game starting with {2}. The reservoir for the next entry (move) is {1, 4}, q = k = 2 from row n = 5 of A348390. If the game continues as {2, 1}, the next move comes form the reservoir for q = 1, that is {2, 3, 4, 5}, but 2 has already been removed, that is, the game continues as {2, 1, 3} or {2, 1, 4} or {2, 1, 5}. The next move comes from reservoir {1} or {2,4} or {1} from q = 3 or 4, or 5, respectively, but these numbers have all to be omitted. Thus the three games have an odd number of moves (namely 3) and the player who starts (player A) wins. The game starting with {2, 4} continues as {2, 4, 1} (from q = 4 with reservoir {1, 2}, but 2 has to be omitted). The next move uses either number 3 or 5 (q = 1, {2, 3, 4, 5}, omitting 2 and 4). Then the game finishes either as {2, 4, 1, 3} or as {2, 4, 1, 5} because the reservoir {1} for q = 3 and also for q = 5 cannot be used. In these two cases the second player (B) wins. Thus there are 3 + 2 = 5 games for start number K = 2 if n = 5 (see also row 5, K = 2 in table A348844).