A Special Number Plate

Last night I noticed an unusual car number plate. It was 432 432.  This is a customised number plates as standard number plates follow an AAA 000 pattern, that is three uppercase letters followed by three digits. Presumably the number 432 was of some significance to the person who purchased the plates. This got me thinking about what is special about the number 432.

432 432

PROPERTY 1

The first property of interest is that it's wedged between two prime numbers: 431 and 433. Thus we have:$$432 = \frac{431+433}{2}$$This qualifies it for membership in OEIS A014574:

A014574

Average of twin prime pairs.

The initial members of this sequence are:

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1608

PROPERTY 2

The next property of interest is that it's the sum of two cubes. Specifically:$$432=6^3+6^3$$This property qualifies it for membership in OEIS A003325:

A003325

Numbers that are the sum of 2 positive cubes.

The initial members of this sequence are:

2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343

PROPERTY 3

The next property is that it's the sum of the totients of the first 37 numbers: $$432=\sum_{n=1}^{37} \phi(n)$$This qualifies it for membership in OEIS A002088:

A002088

Sum of totient function: \( \displaystyle{\text{a}(n) = \sum_{k=1}^n \phi(k) } \)

The initial members of the sequence are:

0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, 530, 542, 584, 604, 628, 650, 696, 712, 754, 774, 806, 830, 882, 900, 940, 964

PROPERTY 4

  

Another property makes it a member of OEIS A033833:

A033833

Highly factorizable numbers: numbers with a record number of proper factorizations.

432 turns out to have 56 possible factors which are:

[2, 2, 2, 2, 3, 3, 3], [2, 2, 2, 2, 3, 9], [2, 2, 2, 2, 27], [2, 2, 2, 3, 3, 6], [2, 2, 2, 3, 18], [2, 2, 2, 6, 9], [2, 2 , 2, 54], [2, 2, 3, 3, 3, 4], [2, 2, 3, 3, 12], [2, 2, 3, 4, 9], [2, 2, 3, 6, 6], [2, 2, 3, 36], [2, 2, 4, 27], [2, 2, 6, 18], [2, 2, 9, 12], [2, 2, 108], [2, 3, 3, 3, 8], [2, 3, 3, 4, 6], [2, 3, 3, 24], [2, 3, 4, 18], [2, 3, 6, 12], [2, 3, 8, 9], [2, 3, 72], [ 2, 4, 6, 9], [2, 4, 54], [2, 6, 6, 6], [2, 6, 36], [2, 8, 27], [2, 9, 4], [2, 12, 18], [2, 216], [3, 3, 3, 4, 4], [3, 3, 3, 16], [3, 3, 4, 12], [3, 3, 6, 8], [3, 3, 48], [3, 4, 4, 9], [3, 4, 6, 6], [3, 4, 36], [3, 6, 24], [3, 8, 18], [3, 9, 16], [3, 12, 12], [3, 144], [4, 4, 27], [4, 6, 18], [4, 9, 12], [4, 108], [6, 6, 12], [6, 8, 9], [6, 72], [8, 54], [9, 48], [12, 36], [16, 27], [18, 24]

It can be noted that 666 makes its appearance since 432 = 2 x 6 x 6 x 6.

The initial members of the sequence are:

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 192, 216, 240, 288, 360, 432, 480, 576, 720, 960, 1080, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160, 25920, 28800, 30240, 34560

PROPERTY 5

Another property of 432 is that it's the difference between the squares of two successive primes. Specifically$$ \begin{align} 432&=109^2-107^2\\&=(109+107) \times (109-107)\\&=216 \times 2\\ &=2^4 \times 3^3 \end{align} $$This qualifies 432 for inclusion in OEIS A069482:

A069482

a(\(n\)) = (prime(\(n\)+1))\(^2\) - (prime(\(n\)))\(^2\)

The initial members of the sequence are:

5, 16, 24, 72, 48, 120, 72, 168, 312, 120, 408, 312, 168, 360, 600, 672, 240, 768, 552, 288, 912, 648, 1032, 1488, 792, 408, 840, 432, 888, 3360, 1032, 1608, 552, 2880, 600, 1848, 1920, 1320, 2040, 2112, 720, 3720, 768, 1560, 792, 4920, 5208, 1800, 912, 1848

I'll stop there as I think I've shown that 432 has at least five interesting properties but of course there are many more. There are actually 4018 entries for this number in the OEIS.

Totient and Non-Totient Numbers

I've written about Euler's Totient Function in an eponymously titled post on March 12th 2019. Today however, my diurnal age number count of 26248 alerted me to the fact that this number was a member of OEIS A333020:

A333020

Starts of runs of 3 consecutive even numbers that are all totient numbers    (A002202).

This begged the question as to what a totient number was and as it turns out that:

A totient number is a value of Euler's totient function: that is, an \(m\) for which there is at least one \(n\) for which \( \phi(n) = m\). The valency or multiplicity of a totient number \(m\) is the number of solutions to this equation. A non-totient is a natural number which is not a totient number. Source.

Apart from 1, every totient number is even. This follows from the fact that every natural number is either prime or a product of primes and the totient of a prime number is one less than the number itself. For example, \( \phi(7)=6\) because 1, 2, 3, 4, 5 and 6 are coprime to it. All primes except 2 are odd and so the totient of all odd primes is even. Let's look at the totient numbers between 1 and 100:

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100 

There are 38 totient numbers between 1 and 100, thus 38%. Between 1 and 1000, there are 291 or about 29% and, between 1 and 10000, there are 2374 or about 24% and so. The percentage decreases as the range is extended. The Wikipedia article provides a formula for approximating this but I won't go into that here.

In my post from June of 2019, I looked at OEIS A001274:

A001274

Numbers \(k\) such that \( \phi(k) = \phi(k+1) \).

Such numbers are relatively infrequent as the following list shows:

1, 3, 15, 104, 164, 194, 255, 495, 584, 975, 2204, 2625, 2834, 3255, 3705, 5186, 5187, 10604, 11715, 13365, 18315, 22935, 25545, 32864, 38804, 39524, 46215, 48704, 49215, 49335, 56864, 57584, 57645, 64004, 65535, 73124, 105524, 107864, 123824, 131144, 164175, 184635, ...

 Apparently, in 1999, the following was proved that:

for every integer \(k \geq 2 \) there is a totient number \(m\) of multiplicity \(k\): that is, for which the equation \( \phi(n) = m\) has exactly \(k\) solutions. However, no number \(m\) is known with multiplicity \(k = 1\). Source.

It's interesting to look at the distribution of these \(k\)s. For values of \(m\) up to one million, we find that \(m\)=241920 has a \(k\) value of 937. Figure 1 shows the distribution:

Figure 1: permalink

As can be seen, \(k\)=937 is quite an outlier. The next highest \(k\) value is 750. This means that up to the one million mark, there are no \(k\) values between 751 to 936 inclusive.

There are many sequences in the OEIS that relate to totient numbers. Here is one example:

A050495

Numbers that are the first term of at least one arithmetic progression with 4 or more terms all having the same value of Euler's totient function \( \phi(x) \).

The example is given of \( \phi(x) \)72 = \( \phi(x) \)(78) = \( \phi(x) \)(84) = \( \phi(x) \)(90) = 24, so 72 is a member of the sequence.

Figure 2 shows the totient numbers between 1 and 72, along with the numbers that produce these totient values:

Figure 2: source