The number associated with my diurnal age today, 27293, has a very interesting property that qualifies it for membership in OEIS A181622 (permalink):
A181622 | Sequence starting with 1 such that the sum of any two distinct terms has three distinct prime factors. |
The sequence begins 1, 29, 41, 281, 401, 1089, 1585, 2289, 4629, 27293 and thus the sum of any two of these numbers produces a sphenic number. There are 45 combinations producing the following sums:
1 + 29 = 30 = 2 * 3 * 5
1 + 41 = 42 = 2 * 3 * 7
1 + 281 = 282 = 2 * 3 * 47
1 + 401 = 402 = 2 * 3 * 67
1 + 1089 = 1090 = 2 * 5 * 109
1 + 1585 = 1586 = 2 * 13 * 61
1 + 2289 = 2290 = 2 * 5 * 229
1 + 4629 = 4630 = 2 * 5 * 463
1 + 27293 = 27294 = 2 * 3 * 4549
29 + 41 = 70 = 2 * 5 * 7
29 + 281 = 310 = 2 * 5 * 31
29 + 401 = 430 = 2 * 5 * 43
29 + 1089 = 1118 = 2 * 13 * 43
29 + 1585 = 1614 = 2 * 3 * 269
29 + 2289 = 2318 = 2 * 19 * 61
29 + 4629 = 4658 = 2 * 17 * 137
29 + 27293 = 27322 = 2 * 19 * 719
41 + 281 = 322 = 2 * 7 * 23
41 + 401 = 442 = 2 * 13 * 17
41 + 1089 = 1130 = 2 * 5 * 113
41 + 1585 = 1626 = 2 * 3 * 271
41 + 2289 = 2330 = 2 * 5 * 233
41 + 4629 = 4670 = 2 * 5 * 467
41 + 27293 = 27334 = 2 * 79 * 173
281 + 401 = 682 = 2 * 11 * 31
281 + 1089 = 1370 = 2 * 5 * 137
281 + 1585 = 1866 = 2 * 3 * 311
281 + 2289 = 2570 = 2 * 5 * 257
281 + 4629 = 4910 = 2 * 5 * 491
281 + 27293 = 27574 = 2 * 17 * 811
401 + 1089 = 1490 = 2 * 5 * 149
401 + 1585 = 1986 = 2 * 3 * 331
401 + 2289 = 2690 = 2 * 5 * 269
401 + 4629 = 5030 = 2 * 5 * 503
401 + 27293 = 27694 = 2 * 61 * 227
1089 + 1585 = 2674 = 2 * 7 * 191
1089 + 2289 = 3378 = 2 * 3 * 563
1089 + 4629 = 5718 = 2 * 3 * 953
1089 + 27293 = 28382 = 2 * 23 * 617
1585 + 2289 = 3874 = 2 * 13 * 149
1585 + 4629 = 6214 = 2 * 13 * 239
1585 + 27293 = 28878 = 2 * 3 * 4813
2289 + 4629 = 6918 = 2 * 3 * 1153
2289 + 27293 = 29582 = 2 * 7 * 2113
4629 + 27293 = 31922 = 2 * 11 * 1451
Only one of the numbers (2289) in this set is sphenic itself and, after the initial 1, the next four numbers are prime:
1 = 1
29 = 29
41 = 41
281 = 281
401 = 401
1089 = 3^2 * 11^2
1585 = 5 * 317
2289 = 3 * 7 * 109
4629 = 3 * 1543
27293 = 7^2 * 557
Beyond 27293, the next numbers in the sequence are 74873, 965813, 2536781, 4479197, 36730306, 150318056 and 4527046433.
It's natural to think about the biprime analog of this sequence and up to 100,000 at least, the members are rather sparse. They are 1, 5, 9, 86 and 212 (permalink). The details are as follows (permalink):
1 = 1
5 = 5
9 = 3^2
86 = 2 * 43
212 = 2^2 * 53
There are 10 combinations of these numbers taken two at a time
1 + 5 = 6 = 2 * 3
1 + 9 = 10 = 2 * 5
1 + 86 = 87 = 3 * 29
1 + 212 = 213 = 3 * 71
5 + 9 = 14 = 2 * 7
5 + 86 = 91 = 7 * 13
5 + 212 = 217 = 7 * 31
9 + 86 = 95 = 5 * 19
9 + 212 = 221 = 13 * 17
86 + 212 = 298 = 2 * 149
Extending to numbers with four distinct prime factors, we get the following initial sequence: 1, 209, 1121, 2989, 11381, 34889, 47701 (permalink). The details are as follows (permalink):
1 = 1
209 = 11 * 19
1121 = 19 * 59
2989 = 7^2 * 61
11381 = 19 * 599
34889 = 139 * 251
47701 = 47701
There are 21 combinations of these numbers taken two at a time
1 + 209 = 210 = 2 * 3 * 5 * 7
1 + 1121 = 1122 = 2 * 3 * 11 * 17
1 + 2989 = 2990 = 2 * 5 * 13 * 23
1 + 11381 = 11382 = 2 * 3 * 7 * 271
1 + 34889 = 34890 = 2 * 3 * 5 * 1163
1 + 47701 = 47702 = 2 * 17 * 23 * 61
209 + 1121 = 1330 = 2 * 5 * 7 * 19
209 + 2989 = 3198 = 2 * 3 * 13 * 41
209 + 11381 = 11590 = 2 * 5 * 19 * 61
209 + 34889 = 35098 = 2 * 7 * 23 * 109
209 + 47701 = 47910 = 2 * 3 * 5 * 1597
1121 + 2989 = 4110 = 2 * 3 * 5 * 137
1121 + 11381 = 12502 = 2 * 7 * 19 * 47
1121 + 34889 = 36010 = 2 * 5 * 13 * 277
1121 + 47701 = 48822 = 2 * 3 * 79 * 103
2989 + 11381 = 14370 = 2 * 3 * 5 * 479
2989 + 34889 = 37878 = 2 * 3 * 59 * 107
2989 + 47701 = 50690 = 2 * 5 * 37 * 137
11381 + 34889 = 46270 = 2 * 5 * 7 * 661
11381 + 47701 = 59082 = 2 * 3 * 43 * 229
34889 + 47701 = 82590 = 2 * 3 * 5 * 2753
Of course, it's not mandatory to start with 1. Suppose in the case of sphenic numbers, we start with 3. The second number must be 27 so that 3 and 27 add to 30, the first sphenic number. When we started with 1, we needed 29 as the second number. After that, we get the following sequence of numbers in the range up to 40,000: 3, 27, 39, 75, 399, 571, 1027, 1387, 3283. The details are as follows:
3 = 3 27 = 3^3 39 = 3 * 13 75 = 3 * 5^2 399 = 3 * 7 * 19 571 = 571 1027 = 13 * 79 1387 = 19 * 73 3283 = 7^2 * 67 There are 36 combinations of these numbers taken two at a time 3 + 27 = 30 = 2 * 3 * 5 3 + 39 = 42 = 2 * 3 * 7 3 + 75 = 78 = 2 * 3 * 13 3 + 399 = 402 = 2 * 3 * 67 3 + 571 = 574 = 2 * 7 * 41 3 + 1027 = 1030 = 2 * 5 * 103 3 + 1387 = 1390 = 2 * 5 * 139 3 + 3283 = 3286 = 2 * 31 * 53 27 + 39 = 66 = 2 * 3 * 11 27 + 75 = 102 = 2 * 3 * 17 27 + 399 = 426 = 2 * 3 * 71 27 + 571 = 598 = 2 * 13 * 23 27 + 1027 = 1054 = 2 * 17 * 31 27 + 1387 = 1414 = 2 * 7 * 101 27 + 3283 = 3310 = 2 * 5 * 331 39 + 75 = 114 = 2 * 3 * 19 39 + 399 = 438 = 2 * 3 * 73 39 + 571 = 610 = 2 * 5 * 61 39 + 1027 = 1066 = 2 * 13 * 41 39 + 1387 = 1426 = 2 * 23 * 31 39 + 3283 = 3322 = 2 * 11 * 151 75 + 399 = 474 = 2 * 3 * 79 75 + 571 = 646 = 2 * 17 * 19 75 + 1027 = 1102 = 2 * 19 * 29 75 + 1387 = 1462 = 2 * 17 * 43 75 + 3283 = 3358 = 2 * 23 * 73 399 + 571 = 970 = 2 * 5 * 97 399 + 1027 = 1426 = 2 * 23 * 31 399 + 1387 = 1786 = 2 * 19 * 47 399 + 3283 = 3682 = 2 * 7 * 263 571 + 1027 = 1598 = 2 * 17 * 47 571 + 1387 = 1958 = 2 * 11 * 89 571 + 3283 = 3854 = 2 * 41 * 47 1027 + 1387 = 2414 = 2 * 17 * 71 1027 + 3283 = 4310 = 2 * 5 * 431 1387 + 3283 = 4670 = 2 * 5 * 467
Obviously, there are many more possibilities to explore here.
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