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

Author

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

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

Positions of records are given in A060274.

Extensions

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