Composites, Prime Factors and Anagrams

Let's consider the prime factors of a composite number. The number associated with my diurnal age today is 27746 so let's take that. It's factorisation is as follows:

$$27746=2 \times 13873$$ The digits making up these factors can be sorted and listed as: $$\text{digits of prime factors }=[1, 2, 3, 3, 7, 8]$$ Now the digits of the number itself can also be sorted and listed as: $$\text{digits of number }=[2, 4, 6, 7, 7]$$ Clearly the two lists are not the same but are there numbers where the two lists are the same? The answer is yes but up to 40,000, there are only four such numbers and they are 1255, 12955, 17482 and 25105 with the following factorisations: $$\begin{align} 1255 &= 5 \times 251 \\12955 &= 5 \times 2591 \\ 17482 &= 2 \times 8741 \\ 25105 &= 5 \times 5021 \end{align}$$ These numbers are the first four terms of OEIS A280928:

A280928   composite numbers having the same digits as their prime factors (with multiplicity), including zero digits.

The initial members of the sequence are (permalink):

1255, 12955, 17482, 25105, 100255, 101299, 105295, 107329, 117067, 124483, 127417, 129595, 132565, 145273, 146137, 149782, 163797, 174082, 174298, 174793, 174982, 250105, 256315, 263155, 295105, 297463, 307183, 325615, 371893, 536539, 687919, 1002955, 1004251, 1012099, 1025095, 1029955

What about composite numbers that have this property but in other bases? Let's consider base 12 for starters. We find that there are 15 numbers in base 10 that have this property when converted to base 12. They are (permalink):

185, 219, 2165, 2402, 3981, 10031, 21349, 21907, 22049, 24199, 27746, 28802, 29919, 31107, 37387

Notice that the number associated with my diurnal age today, 27746, makes an appearance because when we change to base 12 we have:

$$27746_{10}=14082_{12}=2_{12} \times 8041_{12}$$ All of these numbers can be found in OEIS A260055 but there are some additional numbers like 169 because: $$169_{10}=121_{12}=11_{_{12}}^2$$ As can be seen, the exponent is included which I've not allowed.

What about other bases? Let's work our way down from base 10 to base 2 first. There are no numbers in base 9 but in base 8 there are ten terms (permalink):

85, 771, 4369, 4803, 5359, 6805, 7339, 19405, 24433, 36526

As an example, take 85 where we have:

$$85_{8}=125_{8}=5 \times 21_{8}$$ In base 7, there is only one number in the range up to 40,000 (permalink), namely 30057. This is because: $$30057_{10}=153426_7=3_7 \times 61_7 \times 452_7$$ In base 6, there are 45 terms and they are (permalink):

57, 314, 327, 377, 417, 1387, 1417, 1754, 1874, 1934, 1977, 2157, 2355, 2474, 2487, 2517, 2577, 2987, 5597, 8227, 10394, 10474, 10834, 11014, 11229, 11654, 11667, 12317, 12741, 13067, 13117, 13154, 13155, 13427, 14055, 14114, 14417, 14834, 14907, 15117, 15434, 15537, 15929, 15977, 30827

Let's take 57 as an example, where we have:

$$57_{10}=133_7=3_7 \times 31_7$$ There are no numbers in base 5. In base 4 there are 52 terms and they are (permalink):

1135, 1243, 1639, 2167, 4735, 4855, 4939, 5311, 6589, 7003, 7339, 8503, 16735, 17779, 17965, 18079, 18283, 18589, 18654, 18847, 18871, 18895, 18937, 19063, 19255, 20095, 20166, 22471, 22927, 23479, 23659, 24433, 25071, 25467, 26191, 26941, 27019, 27247, 27637, 28149, 28153, 29839, 30147, 31111, 32703, 32721, 33973, 34399, 35299, 36817, 38071, 38863

Let's take 1135 as an example where we have:

$$1135_{10}=101233_4 =11_4 \times 3203_4$$ In base 3, there are 23 terms and they are (permalink):

7847, 8414, 10927, 21299, 22589, 22838, 23294, 23807, 24451, 24458, 24962, 25018, 25214, 25991, 26174, 26201, 27671, 27881, 29141, 29882, 31073, 32389, 38617

Let's take 7847 as an example, where we have:

$$7847_{10}=101202122_3=21_3 \times 201_3 \times 2012_3$$ For base 2, there are 430 terms so I won't list them here. Let's move up from base 10 to base 11 where there is a single term 18193 because (permalink) $$18193_{10}=1273A_{11}=7_{11} \times 21_{11} \times A3_{11}$$ Base 12 has been covered so let's move on to base 13 where there are no numbers and then on to base 14 where there are three terms (permalink):

4119, 16009, 39817

Let's take 4119 as an example where we have:

$$4119_{10}=1703_{14}=3_{14} \times 701_{14}$$ There are no terms in bases 15 and 16 and that is a good place to stop for now.