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A244231 Maximum "digit" value in Semigreedy Catalan Representation of n, A244159.

%I A244231 #22 Jan 14 2015 17:53:59
%S A244231 0,1,1,1,2,1,1,1,1,2,2,2,3,2,1,1,1,1,2,1,1,1,1,2,2,2,3,2,2,2,2,3,3,3,
%T A244231 4,3,2,2,2,2,3,2,1,1,1,1,2,1,1,1,1,2,2,2,3,2,1,1,1,1,2,1,1,1,1,2,2,2,
%U A244231 3,2,2,2,2,3,3,3,4,3,2,2,2,2,3,2,2,2,2,3,3,3,4,3,3,3,3,4,4,4,5,4,3,3,3,3,4,3,2,2,2,2,3,2,2,2,2,3,3,3,4,3,2,2,2,2,3,2,2,2,2,3,3,3,1
%N A244231 Maximum "digit" value in Semigreedy Catalan Representation of n, A244159.
%C A244231 The first value larger than nine occurs at a(33604) = 10. (33604 = A014143(9)). Note that this sequence is not subject to any corruption by decimal representation as A244159 itself is.
%C A244231 A014143 gives records up to that A014143(9) = 33604, but thereafter, the next record occurs at 57317, for which a(57317) = 11, although A014143(10) = 116103. This is explained by that A244159raw(57317) = [2,3,4,5,6,7,8,9,10,11] and A244159raw(116103) = [1,2,3,4,5,6,7,8,9,10,11] (where A244159raw means the underlying representation, before it is maimed by the decimal representation).
%C A244231 This change is explained by the fact that A014143(n-1) + A014138(n) > A000108(n+1) for n = 1..9, but for n >= 10, A014143(n-1) + A014138(n) < A000108(n+1).
%C A244231 For example, although 16808 = 2*C(9) + 3*C(8) + 4*C(7) + 5*C(6) + 6*C(5) + 7*C(4) + 8*C(3) + 9*C(2) + 10*C(1), its representation in A244159 system is 1000000123, as 16808 = 1*C(10) + 1*C(3) + 2*C(2) + 3*C(1).
%H A244231 Antti Karttunen, <a href="/A244231/b244231.txt">Table of n, a(n) for n = 0..58786</a>
%F A244231 For all n, a(A000108(n)) = 1.
%F A244231 For all n >= 1, a(A014138(n)) = 1.
%F A244231 For all n, a(A014143(n)) = n+1.
%o A244231 (Scheme) (define (A244231 n) (if (zero? n) 0 (apply max (vector->list (A244159raw n))))) ;; Code for A244159raw given in A244159.
%Y A244231 Cf. A244159, A244232, A014420, A197433, A244316.
%K A244231 nonn
%O A244231 0,5
%A A244231 _Antti Karttunen_, Jun 25 2014