A060273 - OEIS

A060273

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.

1, 3, 5, 7, 29, 51, 151, 601, 1631, 7159, 19145, 71515, 378701

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OFFSET

1,2

COMMENTS

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.

REFERENCES

C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 376.

C. Pickover, "Wonders of Numbers", Chapter 78, 'Creator Numbers', Oxford University Press, NY, 2001. pp. 187-189, 343-345.

Ken Shirriff, University of California, personal communication.

LINKS

Table of n, a(n) for n=1..13.

Sean A. Irvine, Java program (github)

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Index to sequences related to the complexity of n

EXAMPLE

a(7) = 151 because 151 is the smallest number that requires 7 digits for its expression.

CROSSREFS

Cf. A060274, A260016.

Sequence in context: A058047 A098860 A106920 * A124077 A288891 A358699

Adjacent sequences: A060270 A060271 A060272 * A060274 A060275 A060276

KEYWORD

nonn,base

AUTHOR

Jason Earls, Mar 23 2001

EXTENSIONS

Entry improved by comments from Tim Peters (tim.one(AT)comcast.net), Nov 14 2004

Missing a(1)=1 inserted and existing terms verified by Sean A. Irvine, Nov 06 2022

STATUS

approved