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:
A prime numberis called a-pointer if the next prime number can be obtained adding 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:
A090009 | Begins the earliest length- |
Numberssuch that plus digit sum of and plus digit sum of 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
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
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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