Contribution for OEIS

[Gerald McGarvey]

Richard,

So far, except for the 255 term, your sequence agrees
with sequence A025043 (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.

Gerald
At 05:56 PM 10/30/2005, Richard Guy wrote:
>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.
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