Contribution for OEIS

[Richard Guy]

The sequence of P-positions (previous-
player-winning positions) in the following
nim-like game, played with a heap of beans,
from which a move is to take a prime number
of beans.  Alternatively, can define a(0)=0,
a(1)=1 and a(n) as the least positive integer,
bigger than  a(n-1), for which all
         a(n)-a(k),    -1 < k < n
are composite.  The following was done by hand,
so needs checking:

   0,   1,   9,  10,  25,  34,  35,  49,  55,  85,
  91, 100, 115, 121, 125, 133, 145, 155, 169, 175,
187, 195, 205, 217, 235, 247, 255, 259, 265, 289,
295, 301, 309, 310, ...

Are there infinitely many even members?

The sequence is surprisingly (to me) regular,
considering how it is generated.

Here are two related sequences, but not
recommended for inclusion in OEIS.  a(n)
is the largest (smallest) number of beans
in a winning move from a heap of  n  beans,
with  a(n) = 0  if there is no winning
move, i.e., if  n  is a P-position:

      0   1   2   3   4   5   6   7   8   9
-------------------------------------------
      0,  0,  2,  3,  3,  5,  5,  7,  7,  0,
10   0, 11, 11, 13, 13,  5,  7, 17, 17, 19,
20  19, 11, 13, 23, 23,  0, 17, 17, 19, 19,
30  29, 31, 31, 23,  0,  0, 11, 37, 37, 29,
40  31, 41, 41, 43, 43, 11, 37, 47, 47,  0,
50  41, 41, 43, 53, 53,  0, 47, 47, 23, 59,
60  59, 61, 61, 53, 29, 31, 41, 67, 67, 59,
70  61, 71, 71, 73, 73, 41, 67, 67, 53, 79,
80  79, 71, 73, 83, 83,  0, 61, 53, 79, 89,
90  89,  0, 83, 83, 59, 61, 71, 97, 97, 89

      0,  0,  2,  2,  3,  5,  5,  7,  7,  0,
10   0,  2,  3,  3,  5,  5,  7,  7, 17, 19,
20  11, 11, 13, 13, 23,  0, 17,  2,  3, 19,
30   5, 31,  7, 23,  0,  0,  2,  2,  3,  5,
40   5,  7,  7, 43, 19, 11, 11, 13, 13,  0,
50  41,  2,  3, 19,  5,  0,  7,  2,  3, 59,
60   5, 61,  7, 53, 29, 31, 11, 67, 13, 59,
70  61, 37, 17, 73, 19, 41, 41, 23, 29, 79,
80  31, 47, 47, 83, 29,  0, 31,  2,  3, 79,
90   5,  0,  7,  2,  3, 61,  5, 97,  7, 89

[Warning: these sequences may contain primes
and almost certainly contain errors.]

Of even less interest, perhaps, is the sequence
of nim-values, which may go something like this:

0, 0, 1, 1, 2, 2, 3, 3, 4, 0, 0, 1, 1, 2, 2, 3, 3,
4, 4, 5, 5, 6, 6, 7, 7, 0, 4, 1, 5, 2, 6, 3, 4, 7,
0, 0, 1, 1, 2, 2, 3, 3, 4, 8, ...

R.