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Chen primes
A prime p such that p + 2 is either prime or semiprime. more

The first 600 Chen primes :
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409, 419, 431, 443, 449, 461, 467, 479, 487, 491, 499, 503, 509, 521, 541, 557, 563, 569, 571, 577, 587, 599, 617, 631, 641, 647, 653, 659, 677, 683, 701, 719, 743, 751, 761, 769, 787, 797, 809, 811, 821, 827, 829, 839, 857, 863, 877, 881, 887, 911, 919, 937, 941, 947, 953, 971, 977, 983, 991, 1009, 1019, 1031, 1039, 1049, 1061, 1091, 1097, 1109, 1117, 1151, 1163, 1187, 1193, 1201, 1217, 1229, 1259, 1277, 1283, 1289, 1291, 1297, 1301, 1319, 1327, 1361, 1367, 1381, 1399, 1409, 1427, 1439, 1451, 1459, 1471, 1481, 1487, 1499, 1511, 1553, 1559, 1567, 1583, 1601, 1607, 1619, 1621, 1637, 1667, 1669, 1697, 1709, 1721, 1733, 1759, 1777, 1787, 1801, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1979, 1997, 2003, 2017, 2027, 2029, 2039, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2179, 2207, 2213, 2237, 2243, 2251, 2267, 2269, 2281, 2309, 2333, 2339, 2351, 2357, 2381, 2389, 2393, 2411, 2417, 2441, 2447, 2459, 2467, 2477, 2531, 2543, 2549, 2557, 2579, 2591, 2609, 2621, 2647, 2657, 2659, 2687, 2699, 2711, 2719, 2729, 2731, 2741, 2777, 2789, 2801, 2837, 2843, 2857, 2861, 2879, 2897, 2909, 2927, 2939, 2957, 2963, 2969, 2971, 2999, 3011, 3037, 3041, 3061, 3083, 3089, 3119, 3137, 3167, 3181, 3187, 3191, 3203, 3221, 3251, 3257, 3259, 3271, 3299, 3307, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3457, 3461, 3467, 3491, 3511, 3527, 3539, 3541, 3557, 3559, 3581, 3593, 3637, 3659, 3671, 3677, 3691, 3709, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3847, 3851, 3863, 3881, 3889, 3907, 3917, 3919, 3929, 3947, 3989, 4001, 4007, 4019, 4049, 4091, 4099, 4127, 4133, 4139, 4157, 4211, 4217, 4229, 4241, 4259, 4271, 4283, 4289, 4297, 4337, 4339, 4349, 4357, 4391, 4397, 4409, 4421, 4441, 4447, 4451, 4481, 4517, 4547, 4567, 4591, 4637, 4643, 4649, 4657, 4679, 4703, 4721, 4733, 4787, 4789, 4799, 4801, 4817, 4861, 4871, 4889, 4909, 4931, 4937, 4967, 4969, 4987, 4999, 5009, 5021, 5039, 5051, 5077, 5087, 5099, 5147, 5153, 5167, 5171, 5189, 5197, 5231, 5261, 5279, 5297, 5303, 5309, 5347, 5351, 5381, 5387, 5399, 5417, 5431, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5581, 5639, 5651, 5657, 5669, 5701, 5711, 5737, 5741, 5791, 5801, 5807, 5813, 5849, 5851, 5867, 5879, 5897, 5903, 5939, 5981, 5987, 6007, 6011, 6029, 6047, 6079, 6089, 6101, 6113, 6131, 6143, 6197, 6247, 6257, 6269, 6287, 6299, 6311, 6317, 6329, 6337, 6359, 6421, 6427, 6449, 6481, 6491, 6521, 6551, 6569, 6581, 6607, 6619, 6637, 6659, 6661, 6689, 6701, 6709, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6841, 6863, 6869, 6899, 6911, 6947, 6959, 6971, 6977, 6997, 7001, 7039, 7043, 7069, 7079, 7109, 7121, 7127, 7129, 7151, 7177, 7193, 7211, 7229, 7247, 7253, 7307, 7309, 7321, 7331, 7349, 7417, 7433, 7451, 7457, 7481, 7487, 7499, 7507, 7517, 7529, 7541, 7547, 7559, 7561, 7589, 7591, 7607, 7649, 7669, 7727, 7757, 7793, 7817, 7829, 7853, 7877, 7901, 7907, 7919, 7937, 7949, 8009, 8011, 8059, 8069, 8081, 8087, 8093, 8117, 8147, 8171, 8191, 8219, 8221, 8231, 8291, 8297, 8329, 8369, 8387, 8389, 8429, 8501, 8527, 8537, 8543, 8581, 8597, 8609, 8627, 8663, 8681, 8689, 8707, 8741, 8747, 8779, 8807, 8819, 8837, 8849, 8861, 8887, 8933, 8951, 8969, 8999, 9001, 9011, 9029, 9041, 9067, 9109, 9181, 9199, 9209, 9221, 9227, 9239, 9257.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1762579 values, from 2 to 99999971).

n\r 0  1 
211762578 2 
313133141449264 3 
408811711881407 4 
51544161544328129818544271 5 
603133141101449263 6 
7133528733577633503933568485227335565 7 
80440611144089804405600440509 8 
90156487483283115682648331601482665 9 
10054416111298180105443270544271 10 
11118998418975618995019003519003819009319066719050251531190022

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.