A Special Class of Semiprimes

Let's recall that a semiprime is a number with two, not necessarily distinct, prime factors. The very first semiprime is 4 = 2 x 2 with only one distinct prime factor. The next is 6 = 2 x 3 with two distinct prime factors. Concatenation involves combining the two factors together so that 2 x 2 becomes 22 and 2 x 3 becomes 23. Concatenating the factors of a semiprime with only one distinct prime factor can never produce a prime because of the repetition of digits. Thus 22 is not prime. However, concatenating the factors of a semiprime with two distinct prime factors can produce a prime. 23 is an example. 

However, the factors need not be written is ascending order. We could just as well write 6 = 3 x 2 and in this case concatenating the digits produces 32 which is not a prime number. The first example of a semiprime whose factors can be concatenated either way to produce a prime is 21 because 21 = 3 x 7 giving 37 and 21 = 7 x 3 giving 73. So this is clear enough. Now let's turn our attention to the so-called emiprimes, a semiprime that remains a semiprime when its digits are reversed. The first example of a semiprime that is an emirpimes is 15 because 15 = 3 x 5 and 51 = 3 x 17.

What I want to find is a list of semiprimes with the following properties:

• the semiprime is also an emirpimes
• the semiprime has two distinct prime factors
• the concatenation of the prime factors of the semiprime in ascending order is a prime
• the concatenation of the prime factors of the semiprime in descending order is a prime
• the emirpimes has two distinct prime factors
• the concatenation of the prime factors of the emirpimes in ascending order is a prime
• the concatenation of the prime factors of the emirpimes in descending order is a prime

This is a demanding list of properties for any semiprime and not surprisingly very few satisfy. Here is a list of such numbers, with factorisation, up to 100,000 (Permalink):

3099 = 3 * 1033

9903 = 3 * 3301

10519 = 67 * 157

11707 = 23 * 509

13993 = 7 * 1999

16387 = 7 * 2341

18247 = 71 * 257

19039 = 79 * 241

30607 = 127 * 241

32667 = 3 * 10889

36367 = 41 * 887

38697 = 3 * 12899

39487 = 7 * 5641

39931 = 73 * 547

70603 = 13 * 5431

70711 = 31 * 2281

72247 = 7 * 10321

73099 = 13 * 5623

74227 = 199 * 373

74281 = 59 * 1259

74289 = 3 * 24763

76029 = 3 * 25343

76363 = 7 * 10909

76623 = 3 * 25541

78361 = 23 * 3407

78493 = 53 * 1481

78619 = 29 * 2711

79683 = 3 * 26561

91501 = 37 * 2473

91687 = 277 * 331

92067 = 3 * 30689

93091 = 127 * 733

98247 = 3 * 32749

99037 = 97 * 1021

That are 34 numbers in the range up to 100,00. Here is the list without factorisation of all the semprimes that satisfy up to ONE MILLION (there are 108 of them):

3099, 9903, 10519, 11707, 13993, 16387, 18247, 19039, 30607, 32667, 36367, 38697, 39487, 39931, 70603, 70711, 72247, 73099, 74227, 74281, 74289, 76029, 76363, 76623, 78361, 78493, 78619, 79683, 91501, 91687, 92067, 93091, 98247, 99037, 100437, 101317, 101899, 104529, 108181, 108789, 120553, 126771, 133243, 134797, 137671, 144523, 147061, 149449, 159427, 160741, 168117, 176731, 176767, 177621, 181801, 184033, 197097, 199879, 312817, 322489, 325441, 328459, 330397, 330481, 331783, 337297, 337897, 338977, 342331, 345493, 350569, 355021, 357393, 365863, 368563, 386197, 387133, 393753, 394543, 711861, 713101, 716779, 717469, 718213, 724951, 734001, 767671, 779833, 790791, 791683, 792733, 793033, 797431, 798733, 925401, 944941, 951679, 954823, 964699, 964717, 965053, 976159, 977617, 978991, 984223, 987801, 996469, 998101

Let's one of these, say 99037, to see that it satisfies. Firstly, we note that 73099 is in the list that we know that it's reversal is an emirpimes. Now its factorisation and concatenations lead to two numbers: 971021 and 102197. Testing confirms that both of these numbers are prime. The emirpimes, 73099 factorises to 13 * 5623 that leads to 135623 and 562313. Again, testing reveals both numbers are prime.

So, out of all the semiprimes in the range up to one million there are only 108 that have the properties listed above. So, we have a very special class of semiprimes indeed. It's interesting to note in the distribution that there are no numbers beginning with 2, 4, 5, 6 or 8 which is to be expected. If a semiprime begins with 2, 4, 5, 6 or 8 then its emirpimes will end in 2, 4, 5, 6 or 8 which means that one of the concatenations of its factors must end in 2 or 5, meaning that it can't be prime. Similarly there are no semiprimes that end in a 0, 2, 4, 5, 6 or 8. In short, all semiprimes that satisfy must start and end with 1, 3, 7 or 9.

Figure 1 shows a table of the frequency of the digital roots of such semiprimes (108 of them in the range up to one million):

Figure 1