Repdigits

On May 4th 2022, I turned 26694 days old and one of the properties of 26694 is that is a member of OEIS A167782: numbers that are repdigits with length > 2 in some base. Hmmm, but what base? Looking at the table from Numbers Aplenty, it can be seen that none of the bases from 2 to 16 satisfy (see Table 1).

Table 1

Testing out the bases from 17 to 36 (permalink), it can be seen that again there are no repdigits (see Table 2).

Table 2

The reason for stopping at base 36 is that we have run out of letters of the alphabet and need to resort to an alternative system for representing numbers in higher bases. Now I resorted to trial and error. Base 37 didn’t satisfy but base 38 did. I found that \( 18 \times 38^2+18 \times 38 + 18 = 26694 \) which we can write as \(18.18.18_{36} \). Of course, I got lucky. The number may have been a repunit in a much higher base. However, it’s easy to write a program to handle these higher bases and avoid wasting time on manual calculation (permalink). See Table 3.

Table 3

The table above shows representations for bases up to 60 and if nothing showed up in this range then calculations could be extended until the first number in the triplet reaches zero, in which case the length is equal to 2 and the number does not meet the criterion. For 26694, this occurs at base 164 where we find that \(0 \times 164^2+ 162 \times 164 + 126 = 26694\) which we can represent as \( 162.126_{164} \).

The initial members of OEIS A167782 are: 0, 7, 13, 15, 21, 26, 31, 40, 42, 43, 57, 62, 63, 73, 80, 85, 86, 91, 93, 111, 114, 121, 124, 127, 129, 133, 146, 156, 157, 170, 171, 172, 182, 183, 211, 215, 219, 222, 228, 241, 242, 255, 259, 266, 273, 285, 292, 307, 312, 314, 333, 341, 342, 343, 364, 365, 366. The accompanying comments can be found in the OEIS entry: definition requires “length > 2” because all numbers n > 2 are trivially represented as “11” in base n-1. 0 included at the suggestion of Franklin T. Adams-Watters (and others) as 0 = 000 in any base.

The 10,000th such number in the sequence is 583,744 which means that the percentage of such numbers up to that limit is about 1.713% or a little over 17 per thousand. Let’s not confuse repdigits with repunits:

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers. Wikipedia

All repdigits are multiples of repunits e.g. 666 is a multiple of the repunit 111.