tetrator Wrote:Thanks for asking, Martin. I'm working with a function from N to N whose value at 1 is 2, its value at 2 is 7, and its value at 3 is A(A(A(61,61),A(61,61)), A(A(61,61),A(61,61))).
Hi. I've seen a sequence like that on Robert Munafo's site. Is yours related to Friedman sequences (nonrepeating sequences of different numbers of letters) as described below?
Here's the extract from the relevant page;
Friedman Sequences
In a 1998 paper, Harvey Friedman describes the problem of finding the longest sequence of letters (where there are N allowed letters) such that no subsequence of letters i through 2i occurs anywhere further on in the sequence. For 1 letter the maximum length is 3: AAA. For 2 letters the longest sequence is 11: ABBBAAAAAAA. For 3 letters the longest sequence is very very long, but not infinite.
He then goes on to show how to construct proofs of lower bounds for N-character sequences using certain special (N+1)-character sequences. With help from R. Dougherty, he found a lower bound for the N=3 case, A7198(158386) = ack-rm(7198,158386) = ack(7198,2,158386) = hy(2,7199,158386) = 2(7199)158386, where x(7199)y represents the 7199th hyper operator.
http://www.mrob.com/pub/math/largenum-4.html
It's amazing that a function can jump suddenly like that from mundane numbers to stratospherically huge ones, in fact I think it qualifies as a mathematical *joke. I'd be interested to know what that function is that you're working on, unless you'd rather keep it private.
* Apologies if this is offtopic but I find the expression for the "Look and Say" sequence funny too, or rather the fact that a sequence that's so simple to express needs such a complicated polynomial to express it;
http://en.wikipedia.org/wiki/Look_and_say_sequence