Looking at the factors of my diurnal age today, I was immediately struck by the fact that each factor was a permutation of the digits of the other. To be specific:
26827 = 139 * 193
I immediately speculated as to how common such an occurrence was. To begin my investigation, I had to exclude square semiprimes such as 121 because they qualify trivially. Here is a permalink to the algorithm that I developed to find all such semiprimes up to one million. Below are the semiprimes along with their factorisation:
403 = 13 * 31
1207 = 17 * 71
2701 = 37 * 73
7663 = 79 * 97
14803 = 113 * 131
23701 = 137 * 173
26827 = 139 * 193
34417 = 127 * 271
35143 = 113 * 311
35263 = 179 * 197
40741 = 131 * 311
43429 = 137 * 317
54841 = 173 * 317
62431 = 149 * 419
70027 = 239 * 293
73159 = 149 * 491
75007 = 107 * 701
89647 = 157 * 571
99919 = 163 * 613
101461 = 241 * 421
102853 = 163 * 631
103039 = 167 * 617
103603 = 313 * 331
117907 = 157 * 751
125701 = 337 * 373
127087 = 167 * 761
128701 = 179 * 719
130771 = 251 * 521
140209 = 149 * 941
141643 = 197 * 719
146791 = 181 * 811
150463 = 379 * 397
153211 = 349 * 439
173809 = 179 * 971
174001 = 191 * 911
182881 = 199 * 919
191287 = 197 * 971
197209 = 199 * 991
201379 = 277 * 727
205729 = 419 * 491
212887 = 359 * 593
230701 = 281 * 821
232909 = 283 * 823
246991 = 367 * 673
247021 = 337 * 733
249979 = 457 * 547
257821 = 347 * 743
273409 = 373 * 733
280081 = 379 * 739
293383 = 397 * 739
295501 = 461 * 641
297709 = 463 * 643
302149 = 467 * 647
326371 = 389 * 839
342127 = 359 * 953
355123 = 379 * 937
367639 = 563 * 653
371989 = 397 * 937
374971 = 569 * 659
382387 = 389 * 983
386803 = 613 * 631
394279 = 419 * 941
427729 = 619 * 691
428821 = 571 * 751
436789 = 577 * 757
453613 = 479 * 947
462031 = 491 * 941
469537 = 617 * 761
503059 = 587 * 857
565129 = 593 * 953
589429 = 683 * 863
643063 = 709 * 907
690199 = 787 * 877
692443 = 739 * 937
698149 = 719 * 971
743623 = 769 * 967
778669 = 797 * 977
824737 = 839 * 983
910729 = 919 * 991
There are 79 semiprimes in total. Here is the list (I'm surprised this sequence hasn't made it into the OEIS but it hasn't and I've no intention of submitting it):
403, 1207, 2701, 7663, 14803, 23701, 26827, 34417, 35143, 35263, 40741, 43429, 54841, 62431, 70027, 73159, 75007, 89647, 99919, 101461, 102853, 103039, 103603, 117907, 125701, 127087, 128701, 130771, 140209, 141643, 146791, 150463, 153211, 173809, 174001, 182881, 191287, 197209, 201379, 205729, 212887, 230701, 232909, 246991, 247021, 249979, 257821, 273409, 280081, 293383, 295501, 297709, 302149, 326371, 342127, 355123, 367639, 371989, 374971, 382387, 386803, 394279, 427729, 428821, 436789, 453613, 462031, 469537, 503059, 565129, 589429, 643063, 690199, 692443, 698149, 743623, 778669, 824737, 910729
As can be seen, 26827 is only the seventh such number. I may well be dead before I see the next such semiprime (34417) that will represent my diurnal age on June 26th 2043. If I'm still here I would have passed my 94th birthday.
Another approach to finding these semiprimes would be to test all the primes in a given range, find what permutations of the digits produce primes and multiply the two together. However, this doesn't make for a very efficient algorithm. When I tried it on SageMathCell, the program timed out so the algorithm linked to in the permalink is far more efficient, producing a speedy output.
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