ABA Numbers

The number (\( \textbf{27783} \)) associated with my diurnal age today has the property that it can be expressed as:$$27783=3 \times 21^3$$Numbers like this are called ABA numbers because they can be expressed in the form:$$ \text{AB}^{\text{A}} \text{ for A, B > 1}$$Up to 40,000, the ABA numbers are as follows (link):

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, 1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592, 2738, 2888, 3000, 3042, 3200, 3362, 3528, 3698, 3872, 3993, 4050, 4232, 4374, 4418, 4608, 4802, 5000, 5120, 5184, 5202, 5408, 5618, 5832, 6050, 6272, 6498, 6591, 6728, 6962, 7200, 7442, 7688, 7938, 8192, 8232, 8450, 8712, 8978, 9248, 9522, 9604, 9800, 10082, 10125, 10240, 10368, 10658, 10952, 11250, 11552, 11858, 12168, 12288, 12482, 12800, 13122, 13448, 13778, 14112, 14450, 14739, 14792, 15138, 15309, 15488, 15625, 15842, 16200, 16384, 16562, 16928, 17298, 17496, 17672, 18050, 18432, 18818, 19208, 19602, 20000, 20402, 20577, 20808, 21218, 21632, 22050, 22472, 22528, 22898, 23328, 23762, 24000, 24200, 24576, 24642, 25088, 25538, 25992, 26244, 26450, 26912, 27378, 27783, 27848, 28322, 28800, 29282, 29768, 30258, 30752, 31250, 31752, 31944, 32258, 32768, 33282, 33800, 34322, 34848, 35378, 35912, 36450, 36501, 36992, 37538, 38088, 38642, 38880, 39200, 39762, 40000

648 is the first ABA number with two representations:$$648 =2\times 18^2 = 3\times 6^3$$The smallest number with three such representations is 344373768:$$344373768 = 8 \times 9^8 = 3\times 486^3=2 \times 13122^2$$The smallest Pythagorean triples made of ABA numbers are:$$ (98304, 131072, 163840) \text{ and } (229376, 786432, 819200)$$which correspond to:$$(3 \times 32^3,\, 2\times 256^2,\, 5\times 8^5) \text{ and } (14\times 2^{14},\, 3\times 64^3,\, 2\times 640^2)$$

Pythagorean Triangles With Integer Sides

It's not obvious from the description in the OEIS that the numbers forming the sequence can in fact be with associated with Pythagorean triads and right angled triangles. Today I turned 27475 days old and one of the properties of this number is its membership of OEIS A334542.

A334542

Numbers \(m\) such that \(m^2 = p^2 + k^2\), with \(p\) > 0, where \(p\) = A007954 (\(m\)) = the product of digits of \(m\).

27475 has a product of digits equal to 1960. Let's form a right angles triangle with 27475 as the hypotenuse and 1960 as one of the two sides forming the right angle. Let's call the other side \(x\). The relationship between \(x\), 1960 and 27475 can be expressed as:$$ \begin{align} x^2+1960^2 &= 27475^2\\x^2 &= 27475^2-1960^2\\x &= \sqrt{27475^2-1960^2} \\  &= 27405 \end{align} $$The square root and value of \(x\) just happens to be an integer but this is rarely the case. In general, with \(m\) as the number and \(p\) as its product of digits (with \(p\) > 0), we will have:$$ \begin{align} x^2 &= m^2-p^2 \\ x &= \sqrt{m^2-p^2} \end{align} $$ Here is a permalink to a program that will generate the 16 members of OEIS A334542 sequence up 40000, excluding the trivial single digit numbers, and showing the Pythagorean triads as well. With the Side 1 column being the products of digits and the Hypotenuse column being the sequence numbers, the results are:

Thus the members of the sequence are 58, 85, 375, 666, 1968, 1998, 3578, 3665, 3891, 4658, 4995, 6675, 7735, 18434, 27475 and 28784. Note the appearance of the Number of the Beast, 666, in the sequence.

The same approach can be made using the sum of digits instead of the product of digits. The only results are 17, 25 and 85 with the familiar triads shown below (Side 1 shows the sum of digits --> permalink):

These types of Pythagorean triads have the property that they are self-referencing. The number and its sum of digits or product of the digits form two sides and Pythagoras' Theorem takes care of the remaining side.

As another example, let's investigate whether there are numbers such that the number and its reversal form a Pythagorean Triad. It turns out that there are eight such numbers in the range up to 100,000. Ignoring the numbers formed by the reversals, these are 56, 5265, 5656, 12705, 56056, 55517, 51557 and 59248 and they form the triads (permalink) shown below (Side 2 column shows the numbers and the Hypotenuse column shows their reversals):

Pythagorean Numbers

Of course I knew about Pythagorean triples and even primitive Pythagorean triples but I hadn't heard of Pythagorean numbers. The term emerged when I was researching my daily number, 24570, using the OEIS. This number was paired with 24576 and the smaller followed the larger in sequence A228875: Pairs of Pythagorean numbers differing by 6. This difference is apparently the minimum possible. The sequence started:

24, 30, 54, 60, 210, 216, 330, 336, 480, 486, 540, 546, 720, 726, 750, 756, 1344, 1350, 1710, 1716, 2160, 2166, 8664, 8670, 8970, 8976, 10080, 10086, 10290, 10296, 12144, 12150, 15600, 15606, 18144, 18150, 24570, 24576, 28560, 28566, 30240, 30246, 34650, 34656

This didn't really explain what constituted a Pythagorean number. However, as I discovered here, the definition of such as number is that it is the area of a Pythagorean triangle and primitive Pythagorean number is the area of a primitive Pythagorean triangle. Sequence A009111 provides an ordered list the areas of Pythagorean triangles, effectively providing a list of the initial Pythagorean numbers. Oddly, 24570 turns out to be 294th and 295th in this list. The reason for this will soon become clear.

While I knew that 24570 was a Pythagorean number and thus the area of a Pythagorean triangle, I didn't know the integer sides that comprised such a triangle but it seemed that there were two possible triangles because the number occupied two positions in the list. It took a little fiddling around in WolframAlpha to come up with the numbers.

Thus the triangles were 84, 585, 591 and 180, 273, 327. The number 24570 is not a primitive Pythagorean number because the members of each triplet are divisible by three. The equivalent Pythagorean triplets are 28, 195, 197 and 60, 91, 109.