<html>
<body>
Richard,<br><br>
So far, except for the 255 term, your sequence agrees<br>
with sequence A025043 (<pre>a(n) not of form prime + a(k), k < n)
(it has 253 instead)
It looks like a(n)/n might approach a constant a little over 9.
</pre>Gerald<br>
At 05:56 PM 10/30/2005, Richard Guy wrote:<br>
<blockquote type=cite class=cite cite="">The sequence of P-positions
(previous-<br>
player-winning positions) in the following<br>
nim-like game, played with a heap of beans,<br>
from which a move is to take a prime number<br>
of beans. Alternatively, can define a(0)=0,<br>
a(1)=1 and a(n) as the least positive integer,<br>
bigger than a(n-1), for which all<br>
a(n)-a(k),
-1 < k < n<br>
are composite. The following was done by hand,<br>
so needs checking:<br><br>
0, 1, 9, 10, 25,
34, 35, 49, 55, 85,<br>
91, 100, 115, 121, 125, 133, 145, 155, 169, 175,<br>
187, 195, 205, 217, 235, 247, 255, 259, 265, 289,<br>
295, 301, 309, 310, ...<br><br>
Are there infinitely many even members?<br><br>
The sequence is surprisingly (to me) regular,<br>
considering how it is generated.<br><br>
Here are two related sequences, but not<br>
recommended for inclusion in OEIS. a(n)<br>
is the largest (smallest) number of beans<br>
in a winning move from a heap of n beans,<br>
with a(n) = 0 if there is no winning<br>
move, i.e., if n is a P-position:<br><br>
0 1 2
3 4 5 6 7
8 9<br>
-------------------------------------------<br>
0, 0, 2, 3, 3,
5, 5, 7, 7, 0,<br>
10 0, 11, 11, 13, 13, 5, 7, 17, 17, 19,<br>
20 19, 11, 13, 23, 23, 0, 17, 17, 19, 19,<br>
30 29, 31, 31, 23, 0, 0, 11, 37, 37, 29,<br>
40 31, 41, 41, 43, 43, 11, 37, 47, 47, 0,<br>
50 41, 41, 43, 53, 53, 0, 47, 47, 23, 59,<br>
60 59, 61, 61, 53, 29, 31, 41, 67, 67, 59,<br>
70 61, 71, 71, 73, 73, 41, 67, 67, 53, 79,<br>
80 79, 71, 73, 83, 83, 0, 61, 53, 79, 89,<br>
90 89, 0, 83, 83, 59, 61, 71, 97, 97, 89<br><br>
0, 0, 2, 2, 3,
5, 5, 7, 7, 0,<br>
10 0, 2, 3, 3, 5, 5,
7, 7, 17, 19,<br>
20 11, 11, 13, 13, 23, 0, 17, 2, 3, 19,<br>
30 5, 31, 7, 23, 0, 0, 2,
2, 3, 5,<br>
40 5, 7, 7, 43, 19, 11, 11, 13, 13,
0,<br>
50 41, 2, 3, 19, 5, 0, 7,
2, 3, 59,<br>
60 5, 61, 7, 53, 29, 31, 11, 67, 13, 59,<br>
70 61, 37, 17, 73, 19, 41, 41, 23, 29, 79,<br>
80 31, 47, 47, 83, 29, 0, 31, 2, 3, 79,<br>
90 5, 0, 7, 2, 3, 61, 5,
97, 7, 89<br><br>
[Warning: these sequences may contain primes<br>
and almost certainly contain errors.]<br><br>
Of even less interest, perhaps, is the sequence<br>
of nim-values, which may go something like this:<br><br>
0, 0, 1, 1, 2, 2, 3, 3, 4, 0, 0, 1, 1, 2, 2, 3, 3,<br>
4, 4, 5, 5, 6, 6, 7, 7, 0, 4, 1, 5, 2, 6, 3, 4, 7,<br>
0, 0, 1, 1, 2, 2, 3, 3, 4, 8, ...<br><br>
R.</blockquote></body>
</html>