A306494 Smallest number m such that n*3^m has 2 or more identical adjacent decimal digits.

11, 8, 10, 8, 11, 7, 9, 5, 9, 11, 0, 7, 2, 4, 10, 2, 4, 6, 7, 8, 8, 0, 5, 4, 2, 9, 8, 4, 6, 10, 4, 2, 0, 8, 6, 6, 1, 1, 1, 8, 3, 3, 3, 0, 9, 5, 5, 1, 2, 11, 3, 7, 2, 5, 0, 7, 6, 2, 1, 7, 6, 2, 7, 5, 3, 0, 6, 4, 4, 9, 7, 3, 5, 1, 1, 1, 0, 8, 2, 5, 7, 3, 3, 3, 1

Offset: 1

Author

Chai Wah Wu, Feb 19 2019

Keywords

• nonn
• base

Comments

a(n) is smallest m such that 3^m*n is in the sequence A171901 (or -1 if no such m exists).

0 <= a(n) <= 35 for all n > 0. This is proved by showing that for each 0 < n < 10^9, there is a number m <= 35 such that 3^m*n mod 10^9 has adjacent identical digits. If n > 0 and n == 0 mod 10^9, then clearly a(n) = 0.

Examples

			a(1) = 11 since 3^11 = 177147 has 2 adjacent digits '7' and no smaller power of 3 has adjacent identical digits.
Record values:
a(1) = 11
a(241) = 12
a(2392) = 14
a(35698) = 15
a(267345) = 16
a(893521) = 17
a(29831625) = 18
a(3232453125) = 19
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A171901, A306305.

Programs

Python

def A306494(n):
    m, k= 0, n
    while True:
        s = str(k)
        for i in range(1,len(s)):
            if s[i] == s[i-1]:
                return m
        m += 1
        k *= 3

Formula

a(A171901(n)) = 0.