More About Balanced Numbers

On Friday, the 24th March 2023 (almost a year ago now), I created a post titled Balanced Numbers. I defined these numbers to be those whose sums of digits to the left and right of their centre points or centre digits were equal. Any number with \(2k\) digits where \(k=1,2,3, \dots \) will have a centre point, for example 2314 where:$$ \overbrace{23}^{\text{sum is 5}} \, \overbrace{14}^{\text{sum is 5}}$$Any number with \(2k+1\) digits where \(k=1,2,3, \dots\) will have a centre digit, for example 27018 where$$ \overbrace{27}^{\text{sum is 9}}  0  \overbrace{18}^{\text{sum is 9}}$$There are 2764 such numbers in the range up to 40,000. However, there is a special category of balanced numbers that are far less numerous and these have the additional property that they have an odd number of digits such that the centre digit is equal to the arithmetic digital root of the number. An example is the number associated with my diurnal age today, 27363 with an arithmetic digital root of 3:$$ \overbrace{27}^{\text{sum is 9}}  \underbrace{3}_{\text{root}}   \overbrace{63}^{\text{sum is 9}}$$As would be expected, these numbers comprise about 10% of the total number of balanced numbers and in the range up to 40,000, there are 271 of them. One of them contains three digits: 999. The rest contain five digits and these are:

18109, 18118, 18127, 18136, 18145, 18154, 18163, 18172, 18181, 18190, 18209, 18218, 18227, 18236, 18245, 18254, 18263, 18272, 18281, 18290, 18309, 18318, 18327, 18336, 18345, 18354, 18363, 18372, 18381, 18390, 18409, 18418, 18427, 18436, 18445, 18454, 18463, 18472, 18481, 18490, 18509, 18518, 18527, 18536, 18545, 18554, 18563, 18572, 18581, 18590, 18609, 18618, 18627, 18636, 18645, 18654, 18663, 18672, 18681, 18690, 18709, 18718, 18727, 18736, 18745, 18754, 18763, 18772, 18781, 18790, 18809, 18818, 18827, 18836, 18845, 18854, 18863, 18872, 18881, 18890, 18909, 18918, 18927, 18936, 18945, 18954, 18963, 18972, 18981, 18990, 27109, 27118, 27127, 27136, 27145, 27154, 27163, 27172, 27181, 27190, 27209, 27218, 27227, 27236, 27245, 27254, 27263, 27272, 27281, 27290, 27309, 27318, 27327, 27336, 27345, 27354, 27363, 27372, 27381, 27390, 27409, 27418, 27427, 27436, 27445, 27454, 27463, 27472, 27481, 27490, 27509, 27518, 27527, 27536, 27545, 27554, 27563, 27572, 27581, 27590, 27609, 27618, 27627, 27636, 27645, 27654, 27663, 27672, 27681, 27690, 27709, 27718, 27727, 27736, 27745, 27754, 27763, 27772, 27781, 27790, 27809, 27818, 27827, 27836, 27845, 27854, 27863, 27872, 27881, 27890, 27909, 27918, 27927, 27936, 27945, 27954, 27963, 27972, 27981, 27990, 36109, 36118, 36127, 36136, 36145, 36154, 36163, 36172, 36181, 36190, 36209, 36218, 36227, 36236, 36245, 36254, 36263, 36272, 36281, 36290, 36309, 36318, 36327, 36336, 36345, 36354, 36363, 36372, 36381, 36390, 36409, 36418, 36427, 36436, 36445, 36454, 36463, 36472, 36481, 36490, 36509, 36518, 36527, 36536, 36545, 36554, 36563, 36572, 36581, 36590, 36609, 36618, 36627, 36636, 36645, 36654, 36663, 36672, 36681, 36690, 36709, 36718, 36727, 36736, 36745, 36754, 36763, 36772, 36781, 36790, 36809, 36818, 36827, 36836, 36845, 36854, 36863, 36872, 36881, 36890, 36909, 36918, 36927, 36936, 36945, 36954, 36963, 36972, 36981, 36990 

Notice how the first two digits and last two digits of these numbers are all multiples of 9. This would continue with the next five digit numbers being 45109, 45118, 45127 etc. This sequence is not in the OEIS and I don't intend to propose its inclusion but there is a somewhat related sequence and that is OEIS A240927:

A240927

Positive integers with \(2k\) digits (the first of which is not 0) where the sum of the first \(k\) digits equals the sum of the last \(k\) digits.

The members of this sequence are a subset of the Balanced Numbers that I wrote about in my blog post. It should be noted that my use of the term "balanced number" is peculiar to me although not exclusively so. Figure 1 shows an example of another site that uses the term in the same way:

Figure 1: source

Figure 2 shows another example from Geeks for Geeks:

Figure 2: source