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.

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

1,3

Comments

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

Links

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

Example

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.

Crossrefs

Positions of records are given in A060274.

Sequence in context: A067693 A322006 A173419 * A230697 A322163 A075167

Adjacent sequences: A099050 A099051 A099052 * A099054 A099055 A099056

Keyword

nonn

Author

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

Extensions

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

Status

approved