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):
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