## jueves, 26 de enero de 2012

### Products anagram

Since I was a kid, I have  always been amazed by the fact that when multiplying  four or seven by three, the two products obtained have the same digits but in a different position.
3 x 4 = 12
3 x 7 = 21

Now that I'm a little older, not much, it still surprises me that there are numbers that when they are multiplied by two different numbers, its products are a permutation of each other. Apparently you can find a number for each pair of distinct numbers provided that one of these numbers is not a multiple of ten of the other (ie for n and n * 10 ^ m, there is no number that when multiplied by a specific number, its products are not  anagrams).

Two years ago I published  8 sequences based on these facts in the OEIS. The title of each of these sequences is:  a(n) =smallest number such a(n)*n is an anagram of a(n)* X .

For example the sequence for X  equal to four is :
1782, 62937, 54, 1, 2826, 891, 3, 269, 631, 324, 2718, 4311, 3681, 37, 387, 25974, 4401, 477, 45, 48, 256437, 3393, 37, 26523, 3465, 3252, 3699, 34623, 2922, 27972, 27, 271, 284787, 27324, 25971, 263223, 26973, 25974, 2579247, 2514744     (OEIS A175693)

So:

1782   x 1 =      1782    and    1782 x 4 = 7218
62937 x 2 = 125874    and  62937 x 4 = 251748
54        x 3 =       162     and        54 x 4 = 216
and so on.

Sometimes the same number meets the condition for example:
37 x 13 = 481, 37 x 22 = 814, 37 x 4 = 148

If we write these numbers in a table:

 . 1 2 3 4 5 6 7 8 9 . 1 1 125874 1035 1782 142857 1386 1359 113967 1089 . 2 125874 1 1782 62937 5436 5175 774 891 9 . 3 1035 1782 1 54 36 41958 45 9 345 . 4 1782 62937 54 1 2826 891 3 269631 324 . 5 142857 5436 36 2826 1 9 279 252 2439 . 6 1386 5175 41958 891 9 1 693 27 594 . 7 1359 774 45 3 279 693 1 315 18 . 8 113967 891 9 269631 252 27 315 1 297 . 9 1089 9 345 324 2439 594 18 297 1 . . 389350 202879 45265 338449 154135 50734 3487 385390 5116

we can see that even though the values ​​are different, oddly enough, the sum of the values ​​of number one is an anagram to the sum of the values ​​of the number eight: 389350 - 385390