Concatenating Factors of Sphenic Numbers

The number associated with my diurnal age today, 27047, is a sphenic number and it factorises to 17 x 37 x 43. When these factors are concatenated to form a new number, 173743, then this number is prime. By convention, the concatenation is effected in ascending order but I wondered how the primeness would hold up if all possible orders were considered. The answer is as follows with True meaning is prime and False meaning is not:

173743 True

174337 True

371743 False

374317 True

433717 False

431737 False

So 27047 scores 50% with half the concatenations being prime and half not. The natural follow up was to ask what sphenic numbers hold up 100% of the time. It didn't take too long to construct a SageMath algorithm to determine this in the range up to one million (permalink). There are only 15 such numbers in that range and they are:

3311, 27181, 32153, 41237, 53977, 86507, 110971, 125069, 208579, 256413, 500981, 543337, 853811, 901949, 964481

The factorisations of these numbers are:

3311 = 7 * 11 * 43

27181 = 7 * 11 * 353

32153 = 11 * 37 * 79

41237 = 7 * 43 * 137

53977 = 7 * 11 * 701

86507 = 19 * 29 * 157

110971 = 7 * 83 * 191

125069 = 7 * 17 * 1051

208579 = 7 * 83 * 359

256413 = 3 * 127 * 673

500981 = 13 * 89 * 433

543337 = 17 * 31 * 1031

853811 = 7 * 283 * 431

901949 = 19 * 37 * 1283

964481 = 7 * 211 * 653

Checking with the OEIS, I found that this sequence of numbers is listed as OEIS A180679:

A180679

Numbers with three distinct prime factors which when concatenated in any order form prime numbers.

A longer list of sequence members is provided:

3311, 27181, 32153, 41237, 53977, 86507, 110971, 125069, 208579, 256413, 500981, 543337, 853811, 901949, 964481, 1053787, 1144171, 1197851, 1215731, 1344539, 1385189, 1433659, 1549603, 1674741, 1681547, 1699481, 1973479, 2028181

One could carry out a similar investigation on semiprimes and numbers with four distinct prime factors, although the OEIS comments note that there is no term with four distinct prime factors under \(10^8\). Interestingly, the second number is the sequence, 27181, is coming up very soon in my diurnal age count. It is only 134 days away.