Contribution for OEIS

[Dean Hickerson]

Gerald McGarvey wrote:

> 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.

I think the limit is 4.  Suppose, as seems likely, that there are no even
terms except 10, 34, 100, and 310.  Consider a very large odd number n.
 From a heap of size n we can either move to one of size n-2, or one of
size n-p where p is an odd prime.  For most values of n, all of the numbers
n-10, n-34, n-100, and n-310 are composite, so any move to n-p with p odd
is to an even number which is an N-position.  Hence n is a P-position iff
n-2 is an N-position.  So there will be large blocks of consecutive numbers
for which the P-positions are the numbers == 1 (mod 4) and large blocks where
they're the numbers == 3 (mod 4).  We'll switch from one type to the other
only when there's a winning move to 10, 34, 100, or 310.

Dean Hickerson
dean at math.ucdavis.edu