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 A003037 Smallest number of complexity n: smallest number requiring n 1's to build using +, * and ^. (Formerly M0527) 13
 1, 2, 3, 4, 5, 7, 11, 13, 21, 23, 41, 43, 71, 94, 139, 211, 215, 431, 863, 1437, 1868, 2855, 5737, 8935, 15838, 15839, 54357, 95597, 139117, 233195, 470399, 1228247, 2183791, 4388063, 6945587, 13431919, 32329439, 46551023 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The complexity of an integer n is the least number of 1's needed to represent it using only additions, multiplications, exponentiation and parentheses. This does not allow juxtaposition of 1's to form larger integers, so for example, 2 = 1+1 has complexity 2, but 11 does not (concatenating two 1's is not an allowed operation). The complexity of a number has been defined in several different ways by different authors. See the Index to the OEIS for other definitions. - Jonathan Vos Post, Oct 20 2007 REFERENCES W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS W. A. Beyer, Letter to N. J. A. Sloane, 1980 W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy] EXAMPLE An example (usually nonunique) of the derivation of the first 10 values. a(1) = 1, the number of 1's in "1." a(2) = 2, the number of 1's in "1+1 = 2." a(3) = 3, the number of 1's in "1+1+1 = 3." a(4) = 4, the number of 1's in "1+1+1+1 = 4." a(5) = 5, the number of 1's in "1+1+1+1+1 = 5." a(6) = 7, since there are 6 1's in "((1+1)*(1+1+1))+1 = 7." a(7) = 11, since there are 7 1's in "((1+1+1)^(1+1))+1+1 = eleven." a(8) = 13, since there are 8 1's in "((1+1+1)*(1+1+1+1))+1 = thirteen." a(9) = 21, since there are 9 1's in "((1+1+1)*(((1+1)*(1+1+1))+1) = twenty-one." a(10) = 23, since there are 10 1's in "1+((1+1)*(((1+1+1)^(1+1))+1+1)) = twenty-three." MAPLE xmax:= 5:  # get terms <= 10^xmax C:= {1}: A:= 1: CU:= {1}: for n from 2 do    C[n]:= {seq(seq(seq(op(select(`<=`, [a+b, a*b, `if`(b*ilog10(a) <= xmax, a^b, NULL), `if`(a*ilog10(b) <= xmax, b^a, NULL)]          , 10^xmax)), b=C[n-k]), a=C[k]), k=1..floor(n/2))}          minus CU[n-1];    if C[n] = {} then break fi;    A[n]:= min(C[n]);    CU[n]:= CU[n-1] union C[n]; od: seq(A[i], i=1..n-1); # Robert Israel, Jan 08 2015 CROSSREFS Cf. A025280, A005520, A005245, A005421, A117618. Sequence in context: A174291 A007885 A192586 * A259466 A046420 A108318 Adjacent sequences:  A003034 A003035 A003036 * A003038 A003039 A003040 KEYWORD nonn,nice AUTHOR EXTENSIONS More terms from David W. Wilson, May 15 1997 More terms from Sean A. Irvine, Jan 07 2015 STATUS approved

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Last modified October 16 09:29 EDT 2019. Contains 328056 sequences. (Running on oeis4.)