Multiples of 3 containing at least one 9

[Antreas Hatzipolakis]

 

How many n-digit natural numbers contain the digit 9 at least oncε, and are also multiples of 3?

For n = 3, there are 84 numbers
(Problem by Babis Stergiou in a Greek mathematical forum)

APH

 

[Dave Consiglio]

This sequence has terms:

1, 6, 84, 1056, 12504, 142536, 1582824, 17245416...

Similar to https://oeis.org/A088924

I think terms past a(0) are just A088924 / 3. 

#PYTHON

terms = []
for x in range(10):
    keep = 0
    for y in range(10**x,10**(x+1)):
        if '9' in str(y) and y % 3 == 0:
            keep += 1
    terms.append(keep)
print(keep)

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[M F Hasler]

You should use range(10**k+2, 10**(k+1), 3) and omit check for divisibility. 

Also, the number of these numbers is the number of multiples of 3 with n-1 digits where you can insert (or prefix/apprend) a digit 9 anywhere. The only nontrivial thing is to avoid double counting the numbers where the 9 is inserted to the left or to the right of an existing 9.

-M.

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