Contribution for OEIS

[Richard Guy]

Apologies for

(a) not checking that the sequence was already
in (though there were three other possibly
acceptable sequences)

(b) error in sequence, and for writing
`composite' where I should have written `nonprime'

(c) that I still haven't checked if there's
any connexion with `PRIM' in Winning Ways.

Also thanks for those who set me right and
provided partial answers to some of the implied
questions.             R.

On Mon, 31 Oct 2005, Dean Hickerson wrote:

> 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
>