A060274 -id:A060274 - cp's OEIS frontend

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

Jason Earls, Mar 23 2001

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.

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

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.

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

Cf. A060274, A260016.

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

A099053 a(n) is the smallest number of 1s and 2s that are needed to construct n using any number of +, -, *, ^ signs but not allowing concatenation of digits.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 3, 4, 4, 4, 5, 4, 4, 3, 4, 4, 5, 5, 6, 5, 5, 5, 4, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 6, 6, 7, 6, 7, 6, 6, 6, 6, 5, 5, 5, 6, 6, 6, 5, 6, 6, 7, 6, 7, 6, 6, 5, 5, 4, 5, 5, 6, 6, 7, 6, 6, 5, 6, 6, 6, 6, 6, 6, 5, 5, 4, 5, 5, 6, 6, 7, 7

Offset: 1

Tim Peters (tim.one(AT)comcast.net), Nov 14 2004

nonn

Subexpressions can be grouped as needed; equivalently, any number of parentheses can be used in the expression.
Yet another definition of the complexity of a number.
It can be assumed that no subexpression can be <= 0. The only way to generate a negative value is to take a-b with a < b; taking b-a instead gives the absolute value of this expression. For any further number generated using the negative value, the absolute value of that number is obtainable using the absolute value of the subexpression(s). Generating an intermediate zero is useless. - Franklin T. Adams-Watters, Jul 29 2011

			1 = 1, so has complexity 1.
2 = 2, so has complexity 1.
3 = 1+2, so has complexity 2.
4 = 2+2 = 2*2 = 2^2, so has complexity 2.
5 = 2+1+2, so has complexity 3.
...
16 = 2^2^2, so has complexity 3.

David Consiglio, Jr., Table of n, a(n) for n = 1..90
David Consiglio, Jr., Calculations and upper bounds for values through a(200)
Index to sequences related to the complexity of n

Positions of records are given in A060274.

a(35) onwards from David Consiglio, Jr., Oct 19 2023

cp's OEIS Frontend

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.

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