Perfect Powers

Firstly a reminder. What is a perfect power?

A perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, \(n\) is a perfect power if there exist natural numbers \(m > 1\), and \(k > 1\) such that \(m^k = n\). In this case, \(n\) may be called a perfect \(k\)-th power. If \(k = 2\) or \(k = 3\), then \(n\) is called a perfect square or perfect cube, respectively. Sometimes 1 is also considered a perfect power (\(1^k = 1\) for any \(k\)).

Today I turned 28188 days old and one of the interesting properties of this number, apart from the digit 8 appearing three times, is that its sum of digits (SOD) and product of digits (POD) are both perfect powers. This is because:$$ \begin{align} \text{SOD}(28188) &= 27 = 3^3\\ \text{POD}(28188) &= 1024 = 2^{10 }\end{align}$$This got me thinking as to how many positive integers in the range up to 40000 have this property. Well, it turns out that there are 217 and here they are (permalink):

4, 8, 9, 18, 22, 44, 81, 88, 144, 224, 242, 333, 414, 422, 441, 448, 484, 844, 999, 1124, 1133, 1142, 1177, 1214, 1224, 1241, 1242, 1313, 1331, 1339, 1393, 1412, 1421, 1422, 1555, 1717, 1771, 1888, 1933, 2114, 2124, 2141, 2142, 2214, 2222, 2241, 2248, 2284, 2411, 2412, 2421, 2428, 2482, 2824, 2842, 3113, 3131, 3139, 3193, 3311, 3319, 3391, 3913, 3931, 4112, 4121, 4122, 4211, 4212, 4221, 4228, 4282, 4444, 4822, 5155, 5515, 5551, 7117, 7171, 7711, 8188, 8224, 8242, 8422, 8818, 8881, 8888, 9133, 9313, 9331, 11114, 11124, 11133, 11141, 11142, 11214, 11222, 11241, 11248, 11284, 11313, 11331, 11411, 11412, 11421, 11428, 11482, 11824, 11842, 12114, 12122, 12141, 12148, 12184, 12212, 12221, 12222, 12411, 12418, 12481, 12814, 12841, 12888, 13113, 13131, 13311, 13399, 13939, 13993, 14111, 14112, 14121, 14128, 14182, 14211, 14218, 14281, 14488, 14812, 14821, 14848, 14884, 18124, 18142, 18214, 18241, 18288, 18412, 18421, 18448, 18484, 18828, 18844, 18882, 19339, 19393, 19933, 21114, 21122, 21141, 21148, 21184, 21212, 21221, 21222, 21411, 21418, 21481, 21814, 21841, 21888, 22112, 22121, 22122, 22211, 22212, 22221, 22228, 22282, 22444, 22822, 24111, 24118, 24181, 24244, 24424, 24442, 24811, 28114, 28141, 28188, 28222, 28411, 28818, 28881, 31113, 31131, 31311, 31399, 31939, 31993, 33111, 33199, 33399, 33919, 33939, 33991, 33993, 39139, 39193, 39319, 39339, 39391, 39393, 39913, 39931, 39933

What about numbers whose sums of divisors are perfect powers? In the range up to 40000, there are only 18 and they are (permalink):

3, 7, 21, 31, 81, 93, 127, 217, 381, 400, 651, 889, 2667, 3937, 8191, 11811, 24573, 27559

Take 21 with divisors of 1, 3, 7 and 21 as an example:$$ \sigma(21)=32=2^5$$What about numbers whose totients are perfect powers? Let's recall that:

The totient of a number, denoted by the Euler's totient function \(\phi(n)\), is the count of positive integers less than or equal to \(n\) that are coprime (relatively prime) to \(n\). Two numbers are coprime if their only common factor is 1.

Numbers of this sort are more frequent with there being 119 in the range up to 40000. They are (permalink):

5, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 3855, 4080, 4096, 4112, 4352, 4369, 5120, 5140, 5440, 6144, 6168, 6528, 7680, 7710, 8160, 8192, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 13107, 15360, 15420, 16320, 16384, 16448, 17408, 17476, 20480, 20560, 21760, 21845, 24576, 24672, 26112, 26214, 30720, 30840, 32640, 32768, 32896, 34816, 34952

Let's take 20 as an example. The numbers that are coprime to 20 are 1, 3, 7, 9, 11, 13, 17, 19 and so we have we have a count of 8:$$ \phi(20)=8=2^3$$Next, let's consider the sum of the anti-divisors of a number. First let's recall what an antidivisor is using Gemini to explain in very clear terms:

An anti-divisor is a number that fails to divide a target integer by the largest possible margin. While a regular divisor divides a number evenly with no remainder, an anti-divisor leaves the most unbiased, centered remainder possible. [1, 2, 3]

The concept was first formalized by mathematician Jon Perry. [1]

Mathematical Rules

An integer \(d\) (where \(1 < d < n\)) is an anti-divisor of \(n\) if it satisfies one of the following rules: [1, 2]

• For even values of \(d\): The number \(d\) divides \(2n\) evenly. Mathematically, we have:$$n \bmod d = \dfrac{d}{2}$$
• For odd values of \(d\): The number \(d\) divides \(2n - 1\) or \(2n + 1\) evenly. Mathematically we have  [1, 2]:$$n \bmod d = \dfrac{d-1}{2}  \text{ or } n \bmod d = \dfrac{d+1}{2}$$

Visual Example: Anti-divisors of 20

To find the anti-divisors of \(20\), you look for numbers between \(2\) and \(19\) that fail to divide \(20\), but cleanly divide \(40\) (twice \(20\)) or \(39\)/\(41\). [1]

• 3 is an odd anti-divisor because \(3 \times 13 = 39\), leaving a gap of \(1\) (since \(\frac{3-1}{2} = 1\), and \(20 \bmod 3 = 2\)).
• 8 is an even anti-divisor because \(8 \times 5 = 40\). It divides \(2 \times 20\) perfectly.
• 13 is an odd anti-divisor because \(13 \times 3 = 39\). [1]

Key Facts

• The number 1 is never an anti-divisor, as it evenly divides all integers.
• The number 2 has no anti-divisors.
• Prime numbers have a limited number of anti-divisors based heavily on multiples of \(2n\). [1, 2, 3]

For a complete look at integer sequences and properties, you can explore the The Anti-Divisor - OEIS catalog. [1]

There are 20 numbers in the range up to 40000 that satisfy and these are (permalink):

6, 8, 9, 14, 36, 89, 96, 221, 541, 576, 740, 778, 1854, 2114, 2571, 10277, 13631, 16160, 16389, 39428

Let's take 39428 as an example. The antidivisors of this number are:

3, 5, 7, 8, 15, 21, 35, 105, 751, 2253, 3755, 5257, 11265, 15771, 26285

The sum of these anti-divisors is \(65536 = 2^{16}\) and so it is perfect power.

Further investigation could target:

• the sum of the proper divisors
• the sum of the non-divisors
• the determinant of the circulant matrix
• the arithmetic derivative