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

Offset: 1

Author

Jason Earls, Mar 23 2001

Keywords

• nonn
• base

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.

Examples

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

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

• 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

Crossrefs

Cf. A060274, A260016.

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