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 A274061 Number of 1's required to build n using +, * and concatenation of 1's, where the result of concatenation is interpreted as a binary string. 1
 1, 2, 2, 3, 4, 4, 3, 4, 4, 5, 6, 5, 6, 5, 4, 5, 6, 6, 7, 7, 5, 6, 7, 6, 7, 8, 6, 6, 7, 6, 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 7, 8, 8, 6, 7, 8, 7, 6, 7, 8, 8, 9, 8, 9, 7, 8, 9, 9, 7, 8, 7, 6, 7, 8, 8, 9, 9, 9, 8, 9, 8, 9, 10, 8, 9, 9, 10, 11, 9, 8, 9, 10, 8, 9, 10, 9, 9, 10, 8, 9, 9, 7, 8, 9, 8, 9, 8, 9, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Like A005245, but concatenation of ones is allowed and their results are treated as binary representations of integers. Hence 3 can be represented as 11, 7 as 111 and so on. The largest number with complexity n is 2^n-1 (A000225), the concatenation of n 1's. This follows from (2^m-1)(2^n-1) < 2^(m+n)-1 for m, n >= 1. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..20000 Jeremy Tan, Python program EXAMPLE n . minimal expression . number of 1's 1...1....................1 2...1+1..................2 3...11...................2 4...11+1.................3 5...11+1+1...............4 6...11*(1+1).............4 7...111..................3 8...111+1................4 9...11*11................4 10..11*11+1..............5 11..11*11+1+1............6 12..11*(11+1)............5 13..11*(11+1)+1..........6 14..111*(1+1)............5 15..1111.................4 16..1111+1...............5 17..1111+1+1.............6 18..11*11*(1+1)..........6 19..11*11*(1+1)+1........7 20..(11+1+1)(11+1).......7 21..111*11...............5 MAPLE with(numtheory): a:= proc(n) option remember; (k-> `if`(2^k=n+1, k,       min(seq(a(d)+a(n/d), d=divisors(n) minus {1, n}),           seq(a(i)+a(n-i), i=1..n/2))))(ilog2(n+1))     end: seq(a(n), n=1..120);  # Alois P. Heinz, Jun 09 2016 MATHEMATICA a[n_] := a[n] = Function[k, If[2^k == n+1, k, Min[Table[a[d] + a[n/d], {d, Divisors[n] ~Complement~ {1, n}}], Table[a[i] + a[n-i], {i, 1, n/2}]]]] @ Floor[Log[2, n+1]]; Array[a, 100] (* Jean-François Alcover, Mar 27 2017, after Alois P. Heinz *) CROSSREFS Cf. A005245, A133344. Sequence in context: A325954 A243503 A069581 * A284009 A326846 A243220 Adjacent sequences:  A274058 A274059 A274060 * A274062 A274063 A274064 KEYWORD nonn AUTHOR Jeremy Tan, Jun 08 2016 STATUS approved

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Last modified September 30 23:44 EDT 2020. Contains 337440 sequences. (Running on oeis4.)