An emirp is a prime that remains prime when its digits are reversed. Palindromic primes are excluded as the reversed number must be different from the original. For example, 13 and 17 are both emirps because 31 and 71 are prime. Another way to phrase this is say that:
An emirp is a number with one prime factor (itself) such that its reverse (a different number) also only has one factor.
This definition allows for a generalisation and a special class of numbers arises, namely:
Numbers with \(k\) prime factors, counting multiplicity, such that their reversals are different numbers and also contain \(k\) prime factors, counting multiplicity.
It's easy to set up an algorithm to determine all such numbers in a given range for various values of \(k\). Let's consider the range up to 40000 and \(k=1\). This will generate the emirps. There are 980 of them in the given range so I won't list them here. The sequence members can be found at A006567 or by following this permalink. The first few are 13, 17, 31, 37, 71, 73, 79 and 97.
For \(k=2\), we get the semiprimes. There are an impressive 3450 in the range selected so again I won't show them here but the sequence members can be found at A097393 or by following this permalink. The first few are 15, 26, 39, 49, 51, 58, 62, 85, 93 and 94.
When \(k=3\), we get 2750 numbers that satisfy starting with 117. Here is a permalink that will generate the numbers. Let's use 117 as an example.$$ \begin{align} 117 &= 3 \times 3 \times 13\\711 &= 3 \times 3 \times 79 \end{align} $$When \(k=4\), we get 1302 numbers beginning with 126. Here is a permalink that will generate these numbers. Let's use 126 as an example.$$ \begin{align} 126 &= 2 \times 3 \times 3 \times 7 \\ 621 &= 3 \times 3 \times 3 \times 23 \end{align} $$ For \(k=5\), there are 429 numbers starting with 270. Here is a permalink that will generate the numbers. Let's look at 270.$$ \begin{align} 270 &= 2 \times 3 \times 3 \times 3 \times 5 \\ 72 &= 2 \times 2 \times 2 \times 3 \times 3 \end{align}$$For \(k=6\), there are 103 numbers in the range and so I'll list them. Here is a permalink that will generate the numbers. The first such number is 2576:$$ \begin{align} 2576 &= 2^4 \times 7 \times 23 \\6752 &= 2^5 \times 211 \end{align}$$2576, 2970, 4284, 4356, 4410, 4600, 4698, 4824, 5265, 5625, 6534, 6752, 6900, 8250, 8964, 10710, 10890, 13140, 13986, 16236, 16335, 17577, 18504, 19494, 20286, 20574, 21114, 21150, 21160, 21336, 21492, 21576, 21609, 21900, 21996, 22392, 22770, 22788, 22824, 22869, 23058, 23247, 23250, 23496, 23562, 23580, 23598, 24156, 24660, 24975, 25020, 25092, 25104, 25164, 25245, 25300, 25416, 25434, 25608, 25668, 26163, 26334, 26532, 27060, 27108, 27135, 27192, 27240, 27248, 27270, 27405, 27408, 27468, 27588, 27608, 27636, 27816, 28116, 28215, 28314, 28710, 28890, 29052, 29172, 29322, 29340, 29412, 29580, 29750, 29784, 29835, 29900, 29960, 29984, 32967, 34965, 35775, 35937, 36162, 36990, 37026, 38367, 38934
\(k=7\) generates 25 and this is a small enough number such that we can factorise them all. Here is the permalink to generate numbers.
number factor reverse factor
8820 2^2 * 3^2 * 5 * 7^2 288 2^5 * 3^2
21240 2^3 * 3^2 * 5 * 59 4212 2^2 * 3^4 * 13
21708 2^2 * 3^4 * 67 80712 2^3 * 3^2 * 19 * 59
21780 2^2 * 3^2 * 5 * 11^2 8712 2^3 * 3^2 * 11^2
21920 2^5 * 5 * 137 2912 2^5 * 7 * 13
23280 2^4 * 3 * 5 * 97 8232 2^3 * 3 * 7^3
23472 2^4 * 3^2 * 163 27432 2^3 * 3^3 * 127
23625 3^3 * 5^3 * 7 52632 2^3 * 3^2 * 17 * 43
23800 2^3 * 5^2 * 7 * 17 832 2^6 * 13
25560 2^3 * 3^2 * 5 * 71 6552 2^3 * 3^2 * 7 * 13
25584 2^4 * 3 * 13 * 41 48552 2^3 * 3 * 7 * 17^2
25758 2 * 3^5 * 53 85752 2^3 * 3^3 * 397
26280 2^3 * 3^2 * 5 * 73 8262 2 * 3^5 * 17
27432 2^3 * 3^3 * 127 23472 2^4 * 3^2 * 163
27504 2^4 * 3^2 * 191 40572 2^2 * 3^2 * 7^2 * 23
27888 2^4 * 3 * 7 * 83 88872 2^3 * 3 * 7 * 23^2
27900 2^2 * 3^2 * 5^2 * 31 972 2^2 * 3^5
28836 2^2 * 3^4 * 89 63882 2 * 3^3 * 7 * 13^2
29250 2 * 3^2 * 5^3 * 13 5292 2^2 * 3^3 * 7^2
29403 3^5 * 11^2 30492 2^2 * 3^2 * 7 * 11^2
29736 2^3 * 3^2 * 7 * 59 63792 2^4 * 3^2 * 443
29970 2 * 3^4 * 5 * 37 7992 2^3 * 3^3 * 37
30492 2^2 * 3^2 * 7 * 11^2 29403 3^5 * 11^2
34884 2^2 * 3^3 * 17 * 19 48843 3^6 * 67
36828 2^2 * 3^3 * 11 * 31 82863 3^5 * 11 * 31
number factor reverse factor
16560 2^4 * 3^2 * 5 * 23 6561 3^8
25515 3^6 * 5 * 7 51552 2^5 * 3^2 * 179
27864 2^3 * 3^4 * 43 46872 2^3 * 3^3 * 7 * 31
number factor reverse factor 21168 2^4 * 3^3 * 7^2 86112 2^5 * 3^2 * 13 * 23 23424 2^7 * 3 * 61 42432 2^6 * 3 * 13 * 17 23616 2^6 * 3^2 * 41 61632 2^6 * 3^2 * 107 27456 2^6 * 3 * 11 * 13 65472 2^6 * 3 * 11 * 31
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