Tuesday, 4 June 2024

Remarkable Reversals

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).


The algorithm to generate these numbers is quite flexible and if we change the requirement to eight prime factors with multiplicity then, up to one million, we get 147 numbers as opposed to 43 for nine factors. If we change the requirement to ten prime factors then only 11 numbers satisfy and these are 46848, 217152, 219456, 232848, 257664, 259776, 274104, 276048, 415584, 428736, 846369. For 11 factors we only have two numbers satisfying: 295245 and 426816 while for 12 prime factors there are none.

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