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 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. |
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.
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