Consider the following:$$27456=2^6 \times 3 \times 11 \times 13\\65472=31 \times 11 \times 3 \times 2^6$$As can be seen, if we reverse the order of the digits of 27456, then the resultant number (65472) has, as its prime factors, all the prime factors of 27456 but reversed. Admittedly, 2, 3 and 11 are palindromic but its still a remarkable result, made even more so by the fact that both numbers (27456 and 65472) have exactly nine prime factors (with multiplicity).
Ignoring the reversal itself then, up to one million, there is only one other number with these properties and that is 238656:$$238656 = 2^6 \times 3 \times 11 \times 113\\656832 = 311 \times 11 \times 3 \times 2^6$$However, if we remove the requirement that the prime factors themselves must be reversed and require only that the number and its reverse both have nine prime factors with multiplicity then we have the following numbers up to one million (again only the smaller number and not its larger reversal is included):
21168, 23424, 23616, 27456, 41184, 212256, 213192, 215232, 219072, 230208, 236925, 236928, 238656, 251505, 251748, 253824, 255024, 257856, 259968, 271728, 276696, 276768, 291168, 293328, 299808, 373464, 403056, 403488, 404064, 422208, 424116, 424764, 428928, 441045, 441288, 462384, 472608, 492048, 606096, 610688, 612576, 635328, 804168
These numbers form OEIS A109029 (permalink):
A109029 | Numbers that have exactly nine prime factors counted with multiplicity (A046312) whose digit reversal is different and also has 9 prime factors (with multiplicity). |
No comments:
Post a Comment