Multiplicative and Additive Digital Roots

Even though I've written about multiplicative and arithmetic digital roots in numerous posts, it would seem that I've never addressed the obvious question of how many numbers have identical roots. I was searching for properties of the number associated with my diurnal age (28131) when I noticed the following:$$ \begin{align} 28131 &\rightarrow 2 + 8 + 1+3+1 = 15 \rightarrow 1 + 5 =6 \\ 28131 &\rightarrow 2 \times 8 \times 1 \times 3 \times 1 =48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2 = 6 \end{align}$$It turns out that there are \( \textbf{1085} \) such numbers in the range between 1 and 40000, representing 2.7125% of the range. I won't list all of the numbers here but only those from my diurnal age up to 40000 (permalink):

28131, 28167, 28169, 28176, 28178, 28187, 28196, 28223, 28232, 28311, 28322, 28347, 28374, 28437, 28473, 28617, 28619, 28671, 28691, 28716, 28718, 28734, 28743, 28761, 28781, 28817, 28871, 28916, 28961, 29117, 29126, 29162, 29168, 29171, 29186, 29216, 29261, 29612, 29618, 29621, 29681, 29711, 29816, 29861, 29999, 31113, 31128, 31131, 31139, 31169, 31182, 31193, 31196, 31218, 31227, 31234, 31243, 31272, 31281, 31311, 31319, 31324, 31342, 31344, 31391, 31423, 31432, 31434, 31443, 31619, 31677, 31691, 31722, 31767, 31776, 31778, 31787, 31812, 31821, 31877, 31889, 31898, 31913, 31916, 31931, 31961, 31988, 32118, 32127, 32134, 32143, 32172, 32181, 32217, 32226, 32228, 32262, 32271, 32282, 32314, 32336, 32341, 32363, 32413, 32431, 32478, 32487, 32622, 32633, 32712, 32721, 32748, 32784, 32811, 32822, 32847, 32874, 33111, 33119, 33124, 33142, 33144, 33191, 33214, 33236, 33241, 33263, 33326, 33344, 33362, 33412, 33414, 33421, 33434, 33441, 33443, 33477, 33479, 33497, 33557, 33575, 33623, 33632, 33666, 33747, 33749, 33755, 33774, 33794, 33911, 33947, 33974, 34123, 34132, 34134, 34143, 34213, 34231, 34278, 34287, 34312, 34314, 34321, 34334, 34341, 34343, 34377, 34379, 34397, 34413, 34431, 34433, 34728, 34737, 34739, 34773, 34782, 34793, 34827, 34872, 34937, 34973, 35357, 35375, 35537, 35573, 35735, 35753, 36119, 36177, 36191, 36222, 36233, 36323, 36332, 36366, 36636, 36663, 36717, 36771, 36911, 37122, 37167, 37176, 37178, 37187, 37212, 37221, 37248, 37284, 37347, 37349, 37355, 37374, 37394, 37428, 37437, 37439, 37473, 37482, 37493, 37535, 37553, 37617, 37671, 37716, 37718, 37734, 37743, 37761, 37781, 37817, 37824, 37842, 37871, 37934, 37943, 38112, 38121, 38177, 38189, 38198, 38211, 38222, 38247, 38274, 38427, 38472, 38717, 38724, 38742, 38771, 38819, 38891, 38918, 38981, 39113, 39116, 39131, 39161, 39188, 39311, 39347, 39374, 39437, 39473, 39611, 39734, 39743, 39818, 39881

All permutations of any of these numbers will have multiplicative and arithmetic digital roots that are the same. Putting the digits of 28131 in ascending order, we get 11238. If we only consider numbers whose digits are in ascending order, then in the range up to 40000 there are only \( \textbf{74}\) numbers that qualify. These are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 137, 139, 168, 179, 188, 233, 267, 299, 346, 389, 899, 1124, 1157, 1347, 1355, 1469, 1779, 1788, 2236, 2346, 2348, 2778, 3335, 3779, 11126, 11133, 11148, 11177, 11222, 11238, 11279, 11339, 11369, 11579, 11666, 11677, 11679, 11699, 11999, 12237, 12269, 12334, 12444, 12446, 12678, 12689, 12777, 12788, 13344, 13677, 13778, 13889, 14777, 22236, 22238, 23336, 23478, 29999, 33344, 33477, 33479, 33557, 33666

Permutations of the digits of these numbers will generate the other 1011 (1085 - 74) numbers in the range. These numbers are members of OEIS A064702.

A Multiplicity of Digits: Part 2

A variation on the theme of my previous post, that also involves the multiple occurrence of the same digits, are these numbers that comprise a sequence that I've referenced as S107 in my Bespoken for Sequences database. 

Sphenic numbers containing the digit 3 whose three prime factors also contain the digit 3 and whose additive digital root is 3.

The first example of such a number is 1443 = 3 * 13 * 37 with a digital root of 3. There are 61 such numbers in the range up to 40000. Here is the list (permalink):

1443, 3441, 3657, 3999, 4773, 6357, 8103, 9039, 9453, 11037, 11433, 11937, 11973, 13143, 13197, 13287, 13611, 14313, 15483, 17931, 18093, 20397, 20739, 21423, 21783, 21873, 23907, 23943, 24357, 24753, 26319, 28137, 29739, 30441, 30567, 30783, 31341, 31413, 32097, 32457, 32619, 33267, 34077, 34113, 34131, 34437, 34689, 34707, 34743, 35247, 35697, 36507, 36543, 36741, 36921, 37047, 37407, 38001, 38739, 38847, 39603

A twist on this theme is to consider sphenic numbers that do NOT contain the digit 3. Such numbers could be considered as having a hidden multiplicity of digits because the prevalence of the digit is not immediately obvious. The same could be said of the numbers just mentioned but those cases the repeating digit is overtly visible. Here is the revised criteria:

Sphenic numbers NOT containing the digit 3 whose three prime factors also contain the digit 3 and whose additive digital root is 3.

The first such number is 1209 = 3 * 13 * 31 with a digital root of 3. There are 45 such numbers in the range up to 40000. Here they are (permalink):

1209, 1677, 2847, 4017, 5421, 5727, 6789, 7527, 7797, 8697, 9417, 9579, 12207, 12909, 12927, 14547, 14781, 15159, 15429, 16077, 16491, 16887, 17121, 17949, 17967, 18057, 18147, 20217, 20829, 21027, 21459, 22557, 24609, 24771, 24897, 25077, 26247, 26427, 26841, 27507, 28551, 28587, 28767, 28821, 29109

Rather than sphenic numbers, with three distinct prime factors, we could consider biprimes or numbers with two distinct prime factors. Firstly let's look at numbers with properities as follows:

Biprimes containing the digit 2 whose two prime factors also contain the digit 2 and whose additive digital root is 2. 

There are 73 such numbers in the range up to 40000. The first of these is 254 = 2 * 127 with a digital root of 2. Here they are (permalink):

254, 542, 2558, 2594, 2846, 3242, 4286, 4322, 4502, 4682, 5042, 5267, 5294, 5582, 5942, 8462, 12242, 12422, 12458, 12494, 12854, 14258, 16526, 17246, 17642, 18254, 18929, 19442, 20486, 21458, 22502, 22574, 23483, 23654, 24014, 24041, 24086, 24194, 24482, 24554, 24842, 24914, 25022, 25094, 25166, 25202, 25238, 25274, 25526, 25562, 25598, 25706, 25778, 25814, 25958, 25967, 26498, 26534, 27254, 27623, 28442, 28451, 28586, 28829, 29693, 29846, 30242, 30521, 32546, 33842, 35246, 36254, 39629

Again we can consider the revised criteria:

Biprimes NOT containing the digit 2 whose two prime factors also contain the digit 2 and whose additive digital root is 2. 

There are 53 such numbers in the range up to 40000 with the first being 1046 = 2 * 523 with a digital root of 2. Here are the numbers (permalink):

1046, 1658, 3683, 4034, 4106, 4178, 4358, 4538, 4574, 4754, 4853, 4934, 5006, 5078, 5114, 5186, 5366, 5438, 5834, 5906, 6509, 6518, 7058, 7454, 7859, 10046, 11054, 11846, 13646, 14438, 14474, 15167, 15446, 16418, 17858, 19658, 30854, 31646, 33401, 34058, 34418, 34598, 35687, 35858, 36434, 36506, 36578, 37046, 37091, 37613, 38414, 38846, 39854

SOD Prime Chains

Over the years, I've looked at many forms of prime chains but, as far as I know, not prime chains formed by successively adding the sum of the digits of the prime. What got me thinking about this type of prime chain was the number associated with my diurnal age today: 27043. This number is prime and if we add its sum of digits, we get a new prime. Thus, where SOD stands for Sum Of Digits, we have:$$ \overbrace{27043}^{\text{prime}} + \overbrace{16}^{\text{SOD}}=27059 \text{ which is also prime}$$Because 27059 is the next prime after 27043, it is known as an \( \textbf{a-pointer prime} \) defined by Numbers Aplenty as follows:

A prime number  \(p\) is called a-pointer if the next prime number can be obtained adding  \(p\)  to its sum of digits (here the 'a' stands for additive).

When considering prime chains formed by adding the sum of digits, we are only interested in "prime-ness" and not "a-pointer prime-ness". The earliest example of a prime chain begins with the prime 11. If we add its sum of digits, we get 13 and thus we have a prime chain of length 1: $$\overbrace{11}^{\text{prime}}+\overbrace{2}^{\text{SOD}}=\overbrace{13}^{\text{prime}}$$If we add the sum of digits again we get 17 and thus we have a chain of length 2 namely: $$\overbrace{11}^{\text{prime}}+\overbrace{2}^{\text{SOD}}=\overbrace{13}^{\text{prime}} \text{ and } \overbrace{13}^{\text{prime}}+\overbrace{4}^{\text{SOD}}=\overbrace{17}^{\text{prime}}$$Here, both 11 and 13 are a-pointer primes. We do not get a chain of three primes until 277 where the chain is:$$ 277 \rightarrow 293 \rightarrow 307 \rightarrow 317$$None of these primes are a-pointer primes. The first chain of four occurs with 37783:$$37783 \rightarrow 37811 \rightarrow 37831 \rightarrow 37853 \rightarrow 37879$$The first chain of five occurs with 516493:$$516493 \rightarrow 516521 \rightarrow 516541 \rightarrow 516563 \rightarrow 516589 \rightarrow 516623$$These record chains constitute OEIS A090009:

A090009

Begins the earliest length-\(n\) chain of primes such that any term in the chain equals the previous term increased by the sum of its digits.

The initial members are (permalink - will time out beyond 516493):

2, 11, 11, 277, 37783, 516493, 286330897, 286330897, 56676324799

The progressions for the larger numbers are:

• 286330897 286330943 286330981 286331021 286331047 286331081 286331113 286331141 
• 56676324799 56676324863 56676324919 56676324977 56676325039 56676325091 56676325141 56676325187 56676325243 

In conclusion, we must say that 27043 has the unusual property that its successor, 27044, also produces a prime (27061) when its sum of digits is added. Thus 27059 and 27061 form a pair of twin primes. Given this property of 27043, a new sequence could be formulated as follows:

Numbers \(n\) such that \(n\) plus digit sum of \(n\) and \(n+1\) plus digit sum of \(n+1\) are both prime.

These numbers constitute about 1.335% of numbers in the range up to 40000. This is to be expected since the probability of any number having this property is about 0.1, so two in succession would have a probability of about 0.01. The primes resulting from this process are generally twin primes, although perhaps not exclusively. Numbers ending in 9 such as 299 (with sod = 20) will change to 300 (with sod = 3). However, looking at the output below, there are no numbers ending in 9. Interesting. Triplets are not possible as this would mean three successive primes separated by only a single number. The 534 members up to 40000 are:

10, 13, 34, 52, 58, 91, 94, 100, 103, 127, 142, 166, 181, 184, 217, 232, 256, 271, 295, 304, 340, 412, 418, 451, 508, 583, 610, 631, 787, 811, 814, 838, 1024, 1042, 1048, 1081, 1138, 1222, 1264, 1285, 1312, 1420, 1441, 1465, 1468, 1591, 1597, 1600, 1606, 1648, 1681, 1711, 1771, 1861, 1915, 1933, 1975, 2017, 2071, 2074, 2095, 2104, 2122, 2128, 2230, 2254, 2293, 2302, 2326, 2365, 2638, 2671, 2692, 2701, 2767, 2782, 2947, 2980, 3112, 3154, 3241, 3244, 3283, 3316, 3355, 3373, 3445, 3448, 3514, 3538, 3751, 3796, 3805, 3913, 3976, 3994, 4012, 4036, 4075, 4120, 4144, 4210, 4216, 4231, 4255, 4324, 4411, 4495, 4504, 4528, 4618, 4633, 4696, 4705, 4765, 4780, 4945, 4984, 5002, 5008, 5083, 5221, 5263, 5395, 5404, 5425, 5461, 5482, 5623, 5641, 5827, 5845, 5860, 6073, 6121, 6181, 6253, 6277, 6343, 6433, 6547, 6637, 6670, 6676, 6742, 6760, 6766, 6811, 6850, 6925, 6940, 7114, 7192, 7201, 7285, 7315, 7333, 7441, 7465, 7531, 7537, 7570, 7735, 7930, 7978, 8071, 8215, 8272, 8365, 8413, 8521, 8611, 8788, 8815, 8836, 8944, 8968, 8983, 9001, 9025, 9223, 9262, 9394, 9403, 9421, 9442, 9598, 9607, 9688, 9799, 9910, 9976, 10003, 10027, 10060, 10081, 10132, 10261, 10285, 10318, 10420, 10444, 10483, 10516, 10687, 10843, 10867, 10918, 11050, 11056, 11098, 11107, 11146, 11161, 11341, 11473, 11677, 11695, 11704, 11761, 11815, 11923, 11947, 12028, 12061, 12088, 12151, 12226, 12241, 12358, 12592, 12601, 12796, 12805, 12976, 13210, 13324, 13381, 13657, 13672, 13690, 13696, 13705, 13741, 13813, 13855, 13876, 13984, 14002, 14065, 14311, 14371, 14428, 14533, 14572, 14608, 14845, 15124, 15253, 15271, 15343, 15499, 15562, 15631, 15721, 15946, 16045, 16048, 16171, 16399, 16627, 16666, 16798, 16807, 16954, 16996, 17014, 17167, 17185, 17272, 17365, 17470, 17560, 17635, 17656, 17725, 17764, 17815, 17878, 17893, 17902, 17962, 18025, 18028, 18043, 18115, 18265, 18286, 18517, 18883, 18886, 19057, 19123, 19162, 19186, 19360, 19411, 19450, 19522, 19672, 19726, 19816, 19858, 19963, 19987, 20014, 20140, 20218, 20344, 20431, 20458, 20491, 20500, 20695, 20704, 20728, 20785, 20875, 20962, 20986, 21004, 21007, 21046, 21175, 21310, 21358, 21511, 21538, 21571, 21577, 21592, 21601, 21628, 21718, 21823, 22030, 22075, 22093, 22102, 22144, 22255, 22258, 22348, 22525, 22618, 22675, 22723, 22837, 22942, 23020, 23026, 23044, 23353, 23518, 23608, 23647, 23665, 23884, 24091, 24100, 24163, 24892, 24901, 25021, 25153, 25282, 25285, 25390, 25447, 25555, 25576, 25774, 25825, 25912, 25972, 26092, 26101, 26233, 26656, 26674, 26698, 26707, 26836, 26854, 26926, 27043, 27085, 27223, 27262, 27460, 27511, 27517, 27556, 27664, 27715, 27730, 27886, 27916, 28075, 28090, 28162, 28255, 28327, 28384, 28525, 28546, 28588, 28636, 28726, 28987, 29005, 29113, 29374, 29644, 29734, 29848, 29977, 30004, 30130, 30373, 30448, 30538, 30823, 30847, 31108, 31141, 31165, 31234, 31297, 31306, 31492, 31501, 31525, 31696, 31705, 31708, 31747, 31831, 32020, 32044, 32110, 32131, 32173, 32281, 32311, 32353, 32392, 32401, 32425, 32515, 32590, 32776, 32884, 32917, 32950, 33055, 33163, 33271, 33328, 33565, 33580, 33727, 33745, 33784, 33811, 34021, 34114, 34138, 34192, 34201, 34243, 34282, 34351, 34447, 34480, 34486, 34570, 34627, 34735, 34822, 34825, 34936, 35035, 35257, 35299, 35431, 35566, 35698, 35707, 35815, 35872, 35983, 36001, 36085, 36445, 36511, 36754, 36868, 36910, 36991, 37180, 37321, 37342, 37525, 37546, 37564, 37783, 37963, 38221, 38311, 38425, 38440, 38578, 38626, 38644, 38683, 38887, 39142, 39211, 39217, 39322, 39346, 39478, 39814 

If we impose the restriction that \(n\) must be a prime number, then only 66 numbers qualify. Permalink. These numbers are:

13, 103, 127, 181, 271, 631, 787, 811, 1597, 1861, 1933, 2017, 2293, 2671, 2767, 3373, 4231, 5623, 5641, 5827, 6073, 6121, 6277, 6343, 6547, 6637, 7333, 7537, 8521, 9001, 9403, 9421, 10687, 10867, 11161, 11677, 11923, 12241, 12601, 13381, 14533, 15271, 17167, 18043, 18517, 19963, 20431, 21577, 21601, 22093, 24091, 25153, 25447, 27043, 32173, 32353, 32401, 32917, 33811, 34351, 35257, 35983, 37321, 37783, 37963, 39217

If we impose the restriction that \(n+1\) must be a prime number, then 73 numbers qualify. Permalink. These numbers are:

10, 52, 58, 100, 166, 232, 256, 418, 508, 838, 1048, 1222, 1600, 1606, 2128, 2692, 3448, 3538, 3796, 4012, 4210, 4216, 5002, 5008, 5482, 5860, 6760, 7192, 8272, 8836, 8968, 9688, 10060, 10132, 11056, 12226, 13690, 13696, 13876, 17470, 17656, 17902, 18286, 19162, 19726, 20218, 20962, 22030, 22258, 22348, 22618, 22942, 23020, 23026, 23608, 25390, 25576, 25912, 26698, 26926, 27916, 28162, 28546, 30448, 30538, 31306, 33328, 33580, 34282, 34486, 37180, 37546, 39322

Up to 10 million, no two consecutive prime numbers (that is a pair of twin primes) can produce another pair of twin primes.