Rising To The Challenge

From time to time, numbers arise in the count of my diurnal age that seemingly have no properties that interest me. Always however, with a little research, at least one interesting property emerges. Today's number of 27488 is such a number. Now this number factorises as follows:$$27488 = 2^5 \times 859$$The number has two prime factors, 2 and 859, both of which have digit sums that contain only the digit 2 because 859 has a digit sum of 22.

This got me thinking about what other numbers have prime factors where each factor has a digit sum consisting only of the digit 2. I decided to eliminate prime numbers and numbers that are purely powers of 2. This left with 431 numbers in the range up to 40,000 with the majority of these being even numbers. The list can be viewed here. If we exclude the even numbers, there are only 29 odd numbers and these are (permalink):

121, 1111, 1331, 5489, 8459, 8657, 9449, 9647, 10201, 10637, 12221, 14641, 15389, 16379, 17369, 17567, 18359, 19349, 19547, 20537, 21923, 26279, 29249, 29447, 30239, 30437, 31427, 38159, 39149

The first of these numbers is \(121=11^2\). This got me thinking about digits other than 1 so I set about solving this further challenge. Here is what I discovered:

DIGIT ONE

The algorithm I originally used is easily modified so that similar lists can be generated for the other digits. There are 61 numbers that satisfy and the first is \(841 = 29^2\). The last number in the list is \(39527 = 29^2 \times 47\).

841, 1363, 2209, 2407, 3901, 3973, 5017, 5539, 6439, 6583, 6889, 7627, 8131, 8149, 8977, 9193, 10237, 10669, 11371, 12361, 12847, 13207, 13369, 14359, 14899, 15853, 16591, 18589, 18769, 18841, 20821, 21667, 21829, 23323, 23701, 23809, 24389, 26167, 26311, 26419, 29299, 29551, 29929, 30127, 31099, 31639, 32161, 33043, 33727, 34249, 35293, 36031, 36481, 36769, 37903, 38263, 38497, 38587, 39271, 39469, 39527

DIGIT FOUR

No numbers satisfy when the digit is 3 and 22 satisfy when the digit is 4. An example is \(169 = 13^2\) and \(39013 = 13 \times 3001\). Numbers must composite and these numbers up to 40000 are:

169, 403, 961, 1339, 2197, 2743, 3193, 5239, 6541, 10609, 12493, 13273, 15613, 17407, 21733, 26143, 28561, 29791, 31651, 35659, 37231, 39013

DIGIT FIVE

63 numbers satisfy when the digit is 5 and examples are \(115 =  5 \times 23\) and \(38875 = 5^3 \times 311\). Again no numbers that are purely powers of 5 are allowed and all numbers are composite. These numbers up to 40000 are:

115, 205, 529, 565, 575, 655, 943, 1025, 1555, 1681, 2005, 2599, 2645, 2825, 2875, 3013, 3275, 4633, 4715, 5065, 5125, 5155, 5371, 5515, 6505, 7153, 7775, 8405, 9223, 10015, 10025, 10555, 12167, 12751, 12769, 12995, 13225, 14125, 14375, 14803, 15055, 15065, 16375, 16441, 17161, 20005, 21689, 23165, 23299, 23575, 23713, 25325, 25369, 25625, 25775, 26855, 27575, 29923, 32525, 35143, 35765, 38663, 38875

DIGIT SEVEN

None satisfy when the digit is 6 and 63 satisfy when the digit is 7 and examples are \(301 = 7 \times 43\) and \(36661 =  61 \times 601\).  Again no numbers that are purely powers of 7 are allowed and all numbers are composite. Up to 40000, these numbers are:

301, 427, 1057, 1561, 1687, 1849, 2107, 2191, 2317, 2623, 2947, 2989, 3721, 4207, 6493, 7231, 7357, 7399, 7861, 8491, 8617, 9121, 9211, 9247, 9589, 10363, 10927, 11809, 12943, 13459, 13603, 14233, 14701, 14749, 14791, 14917, 15337, 15421, 15547, 16177, 16219, 18103, 18361, 19093, 20191, 20629, 20923, 21847, 22801, 23107, 25681, 25843, 26047, 28021, 28147, 28777, 29407, 29449, 33673, 35077, 35707, 36391, 36661

DIGIT EIGHT

There are 40 numbers that satisfy when the digit is 8 and examples are \(289 = 17^2\) and \(37621 = 17 \times 2213\). These composite numbers up to 40000 are:

289, 901, 1207, 1819, 2809, 3763, 3961, 4267, 4913, 5041, 5671, 7327, 7597, 8551, 8857, 11449, 11917, 12349, 13303, 15317, 16543, 17821, 18037, 19567, 20519, 20791, 22843, 24931, 25687, 26659, 26857, 27217, 27613, 30601, 30923, 35713, 36397, 36991, 37153, 37621

No numbers arise when the digit is 9. So why are there no numbers arising when the digits 3, 6 and 9 are considered. When a digit sum consists only of 3’s or multiples of 3, then it must be divisible by 3 in the first place. This causes it to break down into smaller prime factors.