One sequence that I was surprised NOT to find in the OEIS was one that involves adding the sum and product of a number's digits to the number itself and then comparing the two results. If the results are different but one is a permutation of the other, then the original number is a member of the sequence. Obviously numbers containing the digit 0 will not qualify as the product of the digits will always be 0. This first number to qualify is 36 where we have:
- sum of digits = 3 + 6 = 9
- number + sum of digits = 45
- product of digits = 3 x 6 = 18
- number + product of digits = 54
- 45 and 54 are permutations of the digits 4 and 5
In the range up to 40,000, there are 80 such numbers and they are (permalink):
36, 156, 438, 1145, 3228, 3348, 3414, 3711, 4314, 4689, 5769, 5949, 6219, 7311, 8343, 9216, 11245, 11257, 11439, 11523, 11558, 11619, 12145, 12512, 12821, 13266, 13512, 14346, 14512, 15123, 15212, 15312, 15412, 15512, 15612, 15712, 15812, 16119, 16236, 16344, 16512, 17512, 18221, 18484, 18512, 18551, 18844, 21145, 21512, 21699, 21821, 22314, 23214, 23238, 24216, 24574, 25112, 25474, 27237, 27369, 27999, 28121, 28233, 29331, 31266, 31512, 31896, 32214, 32238, 33597, 34299, 34461, 34554, 34632, 34776, 35112, 35445, 36216, 37341, 38232
For most of these numbers, the two results of adding the sum and the product of the digits to the number produce permutations with digits that are not identical to the original number. However, there are three numbers where this is indeed the case and these numbers are 5769, 14346 and 27369 (permalink):
- 5769 --> 5796 and 7659
- 14346 --> 14364 and 14634
- 27369 --> 27396 and 29637
These numbers are listed in the OEIS and form the initial members of OEIS A246421:
A246421 | Numbers \(n\) such that (\(n\) + digit sum of \(n\)) and (\(n\) + digit product of \(n\)) are nontrivial permutations of the digits of \(n\). |
All the digit sums and the digit products are multiples of 9. The first members of the sequence are as follows:
5769, 14346, 27369, 41346, 52569, 56925, 94725, 122346, 126135, 129213, 143658, 152469, 154269, 155169, 157914, 162135, 192213, 212346, 216135, 219213, 221346, 236124, 238959, 245925, 261135, 263124, 291213, 326124, 328536, 344925, 361647, 362124, 367425, 368892, 392436, 413658
I would surmise that such a series is finite because as the numbers get larger the size of the product of digits when added to the original number generates numbers with far more digits. There are two associated OEIS sequences to OEIS A246421 and they are:
| A246420 | Numbers \(m\) such that (\(m\) + digit sum of \(m\)) is a permutation of the decimal digits of \(m\). |
Listed below are some of numbers coming up for me in terms of my diurnal age (with 27369 marking the starting point):
27369, 27513, 27558, 27702, 27747, 27891, 27936, 28035, 28224, 28269, 28413, 28458, 28602, 28647, 28836, 29124, 29169, 29313, 29358, 29502, 29547, 29736, 29925, 30123, 30168, 30312, 30357, 30501, 30546, 30735, 30924, 30969, 31023, 31068, 31212, 31257, 31401, 31446, 31635, 31824, 31869, 32112, 32157, 32301, 32346, 32535, 32724, 32769, 32913, 32958, 33012, 33057, 33201, 33246, 33435, 33624, 33669, 33813, 33858, 34101, 34146, 34335, 34524, 34569, 34713, 34758, 34902, 34947, 35001, 35046, 35091, 35235, 35424, 35469, 35613, 35658, 35802, 35847, 36135, 36324, 36369, 36513, 36558, 36702, 36747, 36891, 36936, 37035, 37224, 37269, 37413, 37458, 37602, 37647, 37836, 38124, 38169, 38313, 38358, 38502, 38547, 38736, 38925, 39024, 39069, 39213, 39258, 39402, 39447, 39636, 39780, 39825
A243102 | Numbers \(n\) such that the digits of (\(n\) + product of digits of \(n\)) are a nontrivial permutation of the digits of \(n\). |
Listed below are some of numbers coming up for me in terms of my diurnal age (with 27369 marking the starting point):
27369, 28179, 28195, 29123, 29154, 29213, 29381, 29397, 29873, 31126, 31213, 31235, 31238, 31259, 31354, 31365, 31561, 31925, 32113, 32265, 32286, 32341, 32352, 32492, 32538, 32743, 32793, 33125, 33129, 33142, 33158, 33186, 33248, 33253, 33294, 33455, 33456, 33475, 33558, 33585, 33965, 33967, 34135, 34156, 34167, 34351, 34356, 34526, 34535, 34553, 34563, 34599, 34655, 34951, 35123, 35134, 35165, 35231, 35262, 35267, 35361, 35463, 35616, 35625, 35652, 35673, 35684, 35763, 35794, 35837, 35861, 35974, 36123, 36154, 36178, 36213, 36381, 36722, 36825, 36935, 37168, 37813, 37849, 38143, 38153, 39183, 39251
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