Powerful Numbers Revisited

It's been a long time, May 1st 2016 to be precise, since I wrote about powerful numbers. My post on that date was titled Achilles Numbers, Powerful Number and Perfect Powers. It wasn't a deep or lengthy post as Figure 1 demonstrates:

Figure 1

Today I turned 25947 days old and, in my investigation of this number, I was reminded once again of powerful numbers via Numbers Aplenty. Of this number, the site said:

It is a powerful number, because all its prime factors have an exponent greater than 1 and also an Achilles number because it is not a perfect power.

Clicking on the Numbers Aplenty link to powerful numbers, it says that:

In practice, the set of powerful numbers consists of the number 1 plus all numbers in whose factorisations the primes appears with exponents greater than 1. This set coincides with the set of numbers of the form \(a^2b^3\), for \(a,b \ge 1\). 

The powerful numbers up to 25947 are as follows:

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, 1024, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1331, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1728, 1764, 1800, 1849, 1936, 1944, 2000, 2025, 2048, 2116, 2187, 2197, 2209, 2304, 2312, 2401, 2500, 2592, 2601, 2700, 2704, 2744, 2809, 2888, 2916, 3025, 3087, 3125, 3136, 3200, 3249, 3267, 3364, 3375, 3456, 3481, 3528, 3600, 3721, 3844, 3872, 3888, 3969, 4000, 4096, 4225, 4232, 4356, 4489, 4500, 4563, 4608, 4624, 4761, 4900, 4913, 5000, 5041, 5184, 5292, 5324, 5329, 5400, 5408, 5476, 5488, 5625, 5776, 5832, 5929, 6075, 6084, 6125, 6241, 6272, 6400, 6561, 6724, 6728, 6859, 6889, 6912, 7056, 7200, 7225, 7396, 7569, 7688, 7744, 7776, 7803, 7921, 8000, 8100, 8192, 8281, 8464, 8575, 8649, 8712, 8748, 8788, 8836, 9000, 9025, 9216, 9248, 9261, 9409, 9604, 9747, 9800, 9801, 10000, 10125, 10201, 10368, 10404, 10584, 10609, 10648, 10800, 10816, 10952, 10976, 11025, 11236, 11449, 11552, 11664, 11881, 11907, 11979, 12100, 12167, 12168, 12321, 12348, 12500, 12544, 12769, 12800, 12996, 13068, 13225, 13448, 13456, 13500, 13689, 13824, 13924, 14112, 14161, 14283, 14400, 14641, 14792, 14884, 15125, 15129, 15376, 15488, 15552, 15625, 15876, 16000, 16129, 16200, 16384, 16641, 16807, 16875, 16900, 16928, 17161, 17424, 17496, 17576, 17672, 17689, 17956, 18000, 18225, 18252, 18432, 18496, 18769, 19044, 19208, 19321, 19600, 19652, 19683, 19773, 19881, 20000, 20164, 20449, 20736, 20808, 21025, 21125, 21168, 21296, 21316, 21600, 21609, 21632, 21904, 21952, 22201, 22472, 22500, 22707, 22801, 23104, 23328, 23409, 23716, 24025, 24200, 24300, 24336, 24389, 24500, 24649, 24696, 24964, 25000, 25088, 25281, 25600, 25921, 25947

Some powerful numbers are perfect powers because they can be expressed in the form \(m^k\) for \(m > 1\) and \(k>1\). An example is 9000 that can be expressed as \(30^2\). Powerful numbers that are not perfect powers are referred to as Achilles numbers and my number of the day (25947) is just such a number. This number has the special property that it of the form \(a^2b^3\) where \(a\) and \(b\) are both prime. Powerful numbers of this type form OEIS A143610:

A143610

Numbers of the form \(p^2 \times q^3\), where \(p\) and \(q\) are distinct primes.

Figure 2 shows the code that I wrote to generate this sequence is SageMathCell. There may well be more elegant ways to do this but this is what I came up with:

Figure 2: permalink

The sequence of numbers up to and including 25947 is:

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6125, 6728, 7688, 7803, 8575, 8788, 9747, 10952, 11979, 13448, 14283, 14792, 15125, 17672, 19652, 19773, 21125, 22472, 22707, 25947

Here is some further interesting information about powerful numbers, taken from Numbers Aplenty:

There are infinite pairs of consecutive powerful numbers, the smallest being (8, 9), but Erdös, Mollin, and Walsh conjectured that there are no three consecutive powerful numbers. 

Heath-Brown has shown in 1988 that every sufficiently large natural number is the sum of at most three powerful numbers. Probably the largest number which is not the sum of three powerful numbers is 119. 

The sum of the reciprocals of the powerful numbers converges to:$$ \frac{\zeta(2) \times \zeta(3)}{\zeta(6)} \approx  1.9435964 \dots $$Numbers Aplenty also has some interesting information about Achilles numbers.

There are infinite pairs of consecutive Achilles numbers, the smallest being: \(5425069447 = 7^3 \times 41^2 \times 97^2, 5425069448 = 2^3 \times 26041^2\) 

Actually, Richard B. Stanley has proved that each integer can be expressed in infinite ways as the difference of two coprime Achilles numbers. 

Every number greater that 2370 can be expressed as the sum of Achilles numbers. 

The smallest 3 × 3 magic square of Achilles numbers is shown in Figure 3:

Figure 3

The Achilles numbers, up to 25947, are:

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5292, 5324, 5400, 5408, 5488, 6075, 6125, 6272, 6728, 6912, 7200, 7688, 7803, 8575, 8712, 8748, 8788, 9000, 9248, 9747, 9800, 10125, 10368, 10584, 10800, 10952, 10976, 11552, 11907, 11979, 12168, 12348, 12500, 12800, 13068, 13448, 13500, 14112, 14283, 14792, 15125, 15488, 15552, 16000, 16200, 16875, 16928, 17496, 17672, 18000, 18252, 18432, 19208, 19652, 19773, 20000, 20808, 21125, 21168, 21296, 21600, 21632, 22472, 22707, 23328, 24200, 24300, 24500, 24696, 25000, 25088, 25947

In saying that "Every number greater that 2370 can be expressed as the sum of Achilles numbers", it should be borne in mind that some numbers may require as many as five Achilles numbers to be added together. For example, 3333 = 200 + 392 + 648 + 968 + 1125 without repetition of numbers or, with repetition allowed, the following:

72 + 200 + 968 + 968 + 1125 = 3333
72 + 392 + 392 + 1125 + 1352 = 3333
108 + 500 + 800 + 800 + 1125 = 3333

For 3333, the minimum number required is thus five but sums can be formed using more than this e.g. 72 + 200 + 392 + 392 + 1125 + 1152 = 3333.

Achilles Numbers, Powerful Numbers and Perfect Powers

Today I turned 24500 days old and an analysis of this number revealed that it was an Achilles number in the OEIS listings, specifically OEIS A203663: Achilles number whose double is also an Achilles number. The initial members of this sequence are listed as:

432, 972, 1944, 2000, 2700, 3456, 4500, 5292, 5400, 5488, 8748, 9000, 10584, 10800, 12348, 12500, 13068, 15552, 16000, 17496, 18000, 18252, 21168, 21296, 21600, 24300, 24500, 24696, 25000, 26136 

I'd not heard of an Achilles number before and so I investigated. Wikipedia provides this definition:

An Achilles number is a number that is powerful but not a perfect power. A positive integer \(n\) is a powerful number if, for every prime factor \(p\) of \(n\), \(p^2\) is also a divisor. In other words, every prime factor appears at least squared in the factorisation. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as \(m^k\), where \(m\) and \(k\) are positive integers greater than 1.

Now 24500 factorises to \(2^2 \times 5^3 \times 7^2\) and is thus powerful but not a perfect power. So it is an Achilles number. Multiplication by 2 does not change this because 49000 factorises to \(2^3 \times 5^3 \times 7^2\) and again powerful but not a perfect power.

Here is the Wikipedia definition of 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 \(k\)).