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A099053 a(n) is the smallest number of 1s and 2s that are needed to construct n using any number of +, -, *, ^ signs but not allowing concatenation of digits.

%I A099053 #34 Oct 22 2023 17:04:44
%S A099053 1,1,2,2,3,3,4,3,3,4,4,4,5,4,4,3,4,4,5,5,6,5,5,5,4,5,4,5,5,5,5,4,5,5,
%T A099053 5,4,5,5,6,6,7,6,7,6,6,6,6,5,5,5,6,6,6,5,6,6,7,6,7,6,6,5,5,4,5,5,6,6,
%U A099053 7,6,6,5,6,6,6,6,6,6,5,5,4,5,5,6,6,7,7
%N A099053 a(n) is the smallest number of 1s and 2s that are needed to construct n using any number of +, -, *, ^ signs but not allowing concatenation of digits.
%C A099053 Subexpressions can be grouped as needed; equivalently, any number of parentheses can be used in the expression.
%C A099053 Yet another definition of the complexity of a number.
%C A099053 It can be assumed that no subexpression can be <= 0. The only way to generate a negative value is to take a-b with a < b; taking b-a instead gives the absolute value of this expression. For any further number generated using the negative value, the absolute value of that number is obtainable using the absolute value of the subexpression(s). Generating an intermediate zero is useless. - _Franklin T. Adams-Watters_, Jul 29 2011
%H A099053 David Consiglio, Jr., <a href="/A099053/b099053.txt">Table of n, a(n) for n = 1..90</a>
%H A099053 David Consiglio, Jr., <a href="/A099053/a099053.txt">Calculations and upper bounds for values through a(200)</a>
%H A099053 <a href="/index/Com#complexity">Index to sequences related to the complexity of n</a>
%e A099053 1 = 1, so has complexity 1.
%e A099053 2 = 2, so has complexity 1.
%e A099053 3 = 1+2, so has complexity 2.
%e A099053 4 = 2+2 = 2*2 = 2^2, so has complexity 2.
%e A099053 5 = 2+1+2, so has complexity 3.
%e A099053 ...
%e A099053 16 = 2^2^2, so has complexity 3.
%Y A099053 Positions of records are given in A060274.
%K A099053 nonn
%O A099053 1,3
%A A099053 Tim Peters (tim.one(AT)comcast.net), Nov 14 2004
%E A099053 a(35) onwards from _David Consiglio, Jr._, Oct 19 2023