[seqfan] Hofstadter-Conway $10,000 Sequence

[franktaw at netscape.net]

I just submitted a new sequence:

%S A162598 
1,1,2,1,3,4,2,1,5,6,7,3,8,4,2,1,9,10,11,12,5,13,14,6,15,7,3,16,8,4,2,1,
%T A162598 
17,18,19,20,21,9,22,23,24,10,25,26,11,27,12,5,28,29,13,30,14,6,31,15,7,
%U A162598 
3,32,16,8,4,2,1,33,34,35,36,37,38,17,39,40,41,42,18,43,44,45,19,46,47
%N A162598 Ordinal transform of modified A051135.
%C A162598 This is a fractal sequence.
%C A162598 It appears that each group of 2^k terms starts with 1 and 
ends with the remaining powers of two from 2^k down to 2^1.
%F A162598 Let b(1) = 1, b(n) = A051135(n) for n > 1. Then a(n) is the 
number of b(k) that equal b(n) for 1 <= k <= n: sum( 1, 1<=k<=n and 
a(k)=a(n) ).
%Y A162598 Cf. A004001,A051135.
%K A162598 nonn
%O A162598 1,3

The conjecture here might, if established, throw some light on the 
behavior of the Hofstadter-Conway $10,000 Sequence.

(The ordinal transform is the transform specified by the second 
sentence in the formula line.  As noted in A051135, the modified 
sequence here called b(n) is identical to its lower-trimmed 
subsequence; the ordinal transform of such a sequence is a fractal 
sequence, and vice versa.)

Franklin T. Adams-Watters