189

It's not often that I have a very explicit dream about numbers or a number. Last night I was aware that I was in a long queue with unseen people and a voice announced that we should take note of the number that was displayed on the back of the person in front of us. The number was displayed in large black print and in my case the number was 189. I had been thinking of slipping away and leaving the queue but then I realised that the person behind me would have noted the number displayed on my back and that I was thus a marked man.

That was the extent of the dream, although there is a vague memory of another number which I think was 1094. However, this recollection is more mental and quite different to my close-up and very visual encounter with 189. On waking, I was left wondering what was the significance of this number. Firstly, it should be noted that 189 wasn't MY number. It was the number on the back of the person in front of me in the queue. Given that I was behind this person in a winding line of other people, it might be supposed that my number was 190. However, this is sheer conjecture. Let's return to 189 and explore the properties of this number because this is the number that I saw so clearly and closely in front of me.

Given that I buy tickets in the Australian lottery from time to time, I must confess that my first thought was in connection to gambling rather than the number's possible mystic significance. It turns out that mathematically 189 is a lucky number, lucky in the sense that it survives a savage culling process that eliminates every second, then every third, then even fourth etc. number. Up to 189, the lucky numbers are:

[1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189]

The lottery numbers however, range from 1 to 45 and so how to extract numbers in that range from 189. One obvious way to consider the divisors of the numbers. These are:

[1, 3, 7, 9, 21, 27, 63, 189] 

Of these, it can be seen that 1, 3, 7, 9, 21 and 27 qualify. Normally six or seven numbers must be chosen and so, using divisors, we have six numbers. How might we get seven? We could use a modular approach whereby, once 46 is reached, counting starts again from zero. However, zero is not included in the lottery numbers so it is better to simply subtract 45 from numbers larger than 45. In this way, 46 becomes 1 and 63 becomes 18. For 189, more steps are needed:

189 - 45 = 144 | 144 - 45 = 99 | 99 - 45 = 54 | 54 - 45 = 9

Using this approach, the divisors become [1, 3, 7, 9, 18, 21, 27] with 189 yielding 9 which is already a divisor. Thus we get seven numbers and seven sets of six numbers (with the original six number set marked with an asterisk):

[ 1, 3, 7, 9, 18, 21]

[ 1, 3, 7, 9, 18, 27]

 [1, 3, 7, 9, 21, 27]*

[1, 3, 7, 18, 21, 27]

[1, 3, 9, 18, 21, 27]

[1, 7, 9, 18, 21, 27]

[3, 7, 9, 18, 21, 27]

Perhaps the "best" choice amongst this group is [3, 7, 9, 18, 21, 27] because it doesn't include 1 which is a divisor of every number, not just 189.

Another quite different approach is to consider other special properties of 189, like for instance its membership of OEIS A000930: 

A000930

Narayana's cows sequence: a(0) = a(1) = a(2) = 1                       thereafter a(n) = a(n-1) + a(n-3).

The members of this sequence, up to and including 189, are:

1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189

Let's count backwards from 189 until we reach six and then seven numbers that are eligible lottery numbers. Doing this get:

4, 6, 9, 13, 19, 28, 41

Again we can use these seven numbers to generate seven sets of six numbers:

[4, 6, 9, 13, 19, 28]

[4, 6, 9, 13, 19, 41]

[4, 6, 9, 13, 28, 41]

[4, 6, 9, 19, 28, 41]

[4, 6, 13, 19, 28, 41]

[4, 9, 13, 19, 28, 41]

[6, 9, 13, 19, 28, 41]

While perhaps the seed number 1 should be excluded, there's no reason not to include the 2 and 3 in Narayana's cows sequence. This would yield nine eligible numbers and thus many six and seven sets of numbers. It's simply a matter of choice.

Another important property of 189 is that it can be expressed as a sum of two cubes:$$189=64+125=4^3+5^3$$Numbers that are the sum of two positive cubes form OEIS A003325. How can this be translated into lottery numbers? I'm sure that there are ways but maybe it's time to let go of the lottery angle and just look at the various properties of 189. 

The partition [64, 125] is doubly significant for 189 because 64 can be written as a square (\(8^2\) as well as a cube and so it qualifies for admission to OEIS A055394: numbers that are the sum of a positive square and a positive cube. These numbers are relatively frequent and run:

2, 5, 9, 10, 12, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, 68, 72, 73, 76, 80, 82, 89, 91, 100, 101, 108, 113, 122, 126, 127, 128, 129, 134, 141, 145, 148, 150, 152, 161, 164, 170, 171, 174, 177, 185, 189, ...

189 is also a centered cube number because it is of the form:$$n^3+(n+1)^3$$Here \(n=4\) and these numbers form OEIS A005898.

Another property is that it's an Ulam number. It is the smallest possible, unique sum of two smaller Ulam numbers, 87 and 102. Thus the partition [87, 102] is also significant for 189. The Ulam numbers form OEIS A002858. 

189 is an heptagonal (or 7-gonal) number. These numbers form OEIS A085787  and are of the form:$$ \frac{n \, (5n-3)}{2}$$In the case of 189, we have \(n=9\) and the OEIS sequence runs:

0, 1, 7, 18, 34, 55, 81, 112, 148, 189, ...

189 is also an enneagonal (9-gonal) number. These numbers from OEIS A118277 and are of the form:$$ \frac{n \, (7n-5)}{2}$$The sequence runs:

0, 1, 6, 9, 19, 24, 39, 46, 66, 75, 100, 111, 141, 154, 189, ...

189 is also associated with the prime generating polynomials: \(n^2 \pm n+1\) because \(189^2 + 189+1 = 35911\) and \(189^2 - 189+1=35533\) are both prime and this qualifies the number for admission to OEIS A002384 and OEIS A055494.

At the time of writing, there are 6984 entries in the OEIS for 189 and so I'll have to make a stop.