Contribution for OEIS

[Gerald McGarvey]

There are many more multiples of 5 than multiples of 3 in this
sequence (at least that's how it starts off, with around 4 times
as many multiples of 5).
The trend of a(n)/n towards 9.x breaks around n=100 then
descends slowly.

A related sequence is A025044
a(n) not of form prime - a(k), k < n.
0,1,8,14,20,24,25,26,32,38,44,50,56,62,68,74,80,86,90,92,94,...
Are 1 and 25 the only odd numbers in this sequence?
a(n)/n for this sequence has a roughly similar shaped graph,
what is its limit?
My PARI code for A025044 (I don't think I wrote it well):
m=1000; L = listcreate(m); listput(L,0,1); listput(L,1,2); l=2;
for(n=3,m,x=1;for(i=1,l,if(isprime(n+L[i]),x=0));if(x,listput(L,n,l+1);l=l+1))

Gerald

At 10:47 AM 10/31/2005, Richard Guy wrote:
>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