A066409 Least positive integer not representable using exactly n 9's and the operations +-*/().

1, 2, 1, 4, 13, 22, 33, 103, 195, 381, 934, 1858, 3747, 9166, 31279

Offset

1,2

Comments

This sequence allows fractions as intermediate results; else, a(9) would equal 138. - Michael S. Branicky, Feb 08 2023

Links

Table of n, a(n) for n=1..15.

Joe K. Crump, The 9 Nines

Index entries for similar sequences

Example

a(4)=4 because 4 cannot be expressed with exactly 4 nines and the operations +-*/(). E.g. 1 = 9/9+9-9, 2 = 9/9+9/9, 3 = (9+9+9)/9, but 4 has no such representation.
138 = (((9 - (9 / ((9 + 9) + 9))) * (9 + 9)) - 9) - 9.
265 = ((((9 - (9 / (9 + 9))) + 9) + 9) * ((9 * 9) + 9)) / 9.

Prog

(Python)
from fractions import Fraction
from functools import lru_cache
def a(n):
    @lru_cache()
    def f(m):
        if m == 1: return {9}
        out = set()
        for j in range(1, m//2+1):
            for x in f(j):
                for y in f(m-j):
                    out.update([x + y, x - y, y - x, x * y])
                    if y: out.add(Fraction(x, y))
                    if x: out.add(Fraction(y, x))
        return out
    k, s = 1, f(n)
    while k in s: k += 1
    return k
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Feb 08 2023

Crossrefs

Sequence in context: A225117 A088624 A309878 * A006173 A102055 A232376

Adjacent sequences: A066406 A066407 A066408 * A066410 A066411 A066412

Keyword

more,nonn

Author

Joe K. Crump (joecr(AT)carolina.rr.com), Dec 24 2001

Extensions

Corrected by Leonhard Vogt (leonhard.vogt(AT)gmx.ch), Jan 09 2006

a(11)-a(13) from Michael S. Branicky, Feb 08 2023

a(14)-a(15) from Michael S. Branicky, Aug 03 2023

Status

approved