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%I #22 Nov 06 2022 22:25:48
%S 1,3,5,7,29,51,151,601,1631,7159,19145,71515,378701
%N Hard numbers: a(n) = smallest positive number m with f(m) = n, where f(m) is the smallest number of digits that are needed to construct m using only 1's, 2's and any number of +, -, *, ^ signs, allowing concatenation of the digits.
%C It seems that to obtain this sequence we need to impose two additional rules: 1. x-y is allowed only when x-y > 0 (which also applies to A060274). 2. "Allowing concatenation of the digits" *only* applies to the base digits, 1 and 2, not to the decimal representations of subexpression values. So for example, 13 cannot be obtained via: 13 = 1 concat 3 = 1 concat (1+2) because "3" is not a string consisting solely of 1's or 2's, but can be obtained via: 13 = 11 + 2 = (1 concat 1) + 2. Then the example 151 really does have complexity 7 under this measure.
%D C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 376.
%D C. Pickover, "Wonders of Numbers", Chapter 78, 'Creator Numbers', Oxford University Press, NY, 2001. pp. 187-189, 343-345.
%D Ken Shirriff, University of California, personal communication.
%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a060/A060273.java">Java program</a> (github)
%H C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&format=complete">Zentralblatt review</a>
%H <a href="/index/Com#complexity">Index to sequences related to the complexity of n</a>
%e a(7) = 151 because 151 is the smallest number that requires 7 digits for its expression.
%Y Cf. A060274, A260016.
%K nonn,base
%O 1,2
%A _Jason Earls_, Mar 23 2001
%E Entry improved by comments from Tim Peters (tim.one(AT)comcast.net), Nov 14 2004
%E Missing a(1)=1 inserted and existing terms verified by _Sean A. Irvine_, Nov 06 2022
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