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A086833 Minimum number of different addends occurring in any shortest addition chain of Brauer type for a given n, or 0 if n has no shortest addition chain of Brauer type.

%I A086833 #18 Apr 25 2025 09:49:27
%S A086833 1,1,1,2,2,2,2,3,2,3,3,3,3,3,3,4,4,3,3,4,3,4,5,4,4,4,3,4,4,4,4,5,5,5,
%T A086833 4,4,4,4,4,5,5,4,6,5,4,6,4,5,5,5,5,5,5,4,4,5,4,5,5,5,5,5,4,6,6,6,6,6,
%U A086833 6,5,5,5,5,5,5,5,7,5,5,6,4,6,7,5,6,7,5,6,6,5,5,7,5,5,5,6,6,6,6,6,6,6,6,6,5
%N A086833 Minimum number of different addends occurring in any shortest addition chain of Brauer type for a given n, or 0 if n has no shortest addition chain of Brauer type.
%C A086833 n = 12509 is the first n for which a(n) = 0 because it is the smallest number that has no shortest addition chain of Brauer type. - _Hugo Pfoertner_, Jun 10 2006 [Edited by _Pontus von Brömssen_, Apr 25 2025]
%H A086833 Giovanni Resta, <a href="http://www.numbersaplenty.com/ac/">Tables of Shortest Addition Chains</a>, computed by David W. Wilson.
%H A086833 <a href="/index/Com#complexity">Index to sequences related to the complexity of n</a>
%F A086833 a(n) = 0 if and only if n is in A349044. - _Pontus von Brömssen_, Apr 25 2025
%e A086833 a(23)=5 because 23=1+1+2+1+4+9+5 is the shortest addition chain for 23.
%e A086833 For n=9 there are A079301(9)=3 different shortest addition chains, all of Brauer type:
%e A086833 [1 2 3 6 9] -> 9=1+1+1+3+3 -> 2 different addends {1,3}
%e A086833 [1 2 4 5 9] -> 9=1+1+2+1+4 -> 3 different addends {1,2,4}
%e A086833 [1 2 4 8 9] -> 9=1+1+2+4+1 -> 3 different addends {1,2,4}
%e A086833 The minimum number of different addends is 2, therefore a(9)=2.
%Y A086833 Cf. A003064, A003065, A003313, A005766, A008057, A008928, A008933, A079300, A079300, A079301, A349044.
%K A086833 nonn
%O A086833 1,4
%A A086833 _Tatsuru Murai_, Aug 08 2003
%E A086833 Edited by _Hugo Pfoertner_, Jun 10 2006
%E A086833 Escape clause added by _Pontus von Brömssen_, Apr 25 2025