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. |
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