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weak primes
A prime smaller than the average of the two surrounding primes. more

The first 600 weak primes :
3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199, 229, 233, 241, 271, 283, 293, 313, 317, 337, 349, 353, 359, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523, 547, 571, 577, 601, 619, 643, 647, 661, 677, 683, 691, 709, 743, 761, 773, 797, 811, 823, 829, 839, 859, 863, 883, 887, 911, 919, 941, 953, 971, 983, 997, 1013, 1021, 1033, 1039, 1051, 1063, 1069, 1093, 1097, 1109, 1129, 1153, 1171, 1193, 1201, 1217, 1231, 1237, 1259, 1279, 1283, 1291, 1303, 1307, 1321, 1327, 1373, 1381, 1409, 1429, 1433, 1439, 1453, 1459, 1483, 1489, 1493, 1499, 1531, 1553, 1559, 1571, 1583, 1601, 1609, 1613, 1621, 1627, 1637, 1669, 1699, 1709, 1723, 1759, 1789, 1811, 1831, 1873, 1879, 1889, 1913, 1933, 1951, 1979, 1999, 2003, 2017, 2029, 2039, 2069, 2083, 2089, 2099, 2113, 2131, 2143, 2161, 2179, 2207, 2213, 2221, 2239, 2243, 2251, 2269, 2273, 2297, 2311, 2341, 2351, 2357, 2383, 2393, 2399, 2423, 2441, 2447, 2477, 2543, 2551, 2557, 2593, 2621, 2633, 2659, 2663, 2689, 2693, 2699, 2713, 2719, 2731, 2753, 2777, 2791, 2803, 2837, 2843, 2861, 2887, 2909, 2917, 2927, 2939, 2957, 2971, 3001, 3023, 3041, 3049, 3067, 3083, 3089, 3121, 3137, 3169, 3191, 3209, 3221, 3229, 3253, 3259, 3271, 3301, 3323, 3331, 3347, 3361, 3373, 3391, 3413, 3463, 3469, 3499, 3517, 3529, 3533, 3541, 3547, 3559, 3583, 3593, 3617, 3623, 3643, 3673, 3677, 3701, 3709, 3739, 3769, 3779, 3797, 3803, 3823, 3833, 3853, 3863, 3881, 3889, 3911, 3919, 3923, 3931, 3947, 3967, 4003, 4007, 4021, 4027, 4051, 4057, 4079, 4093, 4099, 4111, 4129, 4133, 4139, 4159, 4177, 4219, 4231, 4243, 4261, 4273, 4289, 4297, 4339, 4363, 4373, 4397, 4423, 4451, 4463, 4483, 4493, 4519, 4523, 4549, 4567, 4603, 4639, 4643, 4651, 4663, 4679, 4703, 4723, 4733, 4759, 4789, 4793, 4801, 4817, 4831, 4877, 4889, 4909, 4919, 4933, 4937, 4943, 4957, 4969, 4973, 5003, 5011, 5023, 5059, 5081, 5087, 5101, 5119, 5153, 5171, 5179, 5197, 5209, 5233, 5237, 5281, 5309, 5333, 5351, 5399, 5419, 5443, 5449, 5479, 5483, 5503, 5507, 5521, 5531, 5573, 5581, 5591, 5641, 5653, 5659, 5669, 5693, 5701, 5717, 5743, 5749, 5783, 5791, 5813, 5827, 5843, 5851, 5861, 5869, 5881, 5903, 5927, 5939, 5953, 5987, 6011, 6047, 6053, 6079, 6091, 6101, 6121, 6133, 6151, 6173, 6199, 6203, 6221, 6229, 6271, 6277, 6287, 6301, 6329, 6343, 6361, 6379, 6397, 6427, 6451, 6473, 6481, 6491, 6529, 6553, 6571, 6581, 6607, 6619, 6661, 6679, 6691, 6703, 6709, 6719, 6737, 6763, 6781, 6793, 6803, 6829, 6833, 6841, 6871, 6883, 6911, 6917, 6949, 6961, 6971, 6983, 7001, 7019, 7027, 7043, 7079, 7109, 7129, 7159, 7193, 7213, 7219, 7247, 7253, 7309, 7333, 7351, 7369, 7417, 7433, 7459, 7481, 7489, 7507, 7529, 7541, 7549, 7561, 7577, 7591, 7607, 7621, 7643, 7649, 7673, 7691, 7703, 7727, 7759, 7793, 7829, 7853, 7879, 7883, 7907, 7937, 7951, 7963, 8011, 8017, 8059, 8069, 8089, 8093, 8101, 8123, 8171, 8179, 8191, 8221, 8233, 8237, 8243, 8273, 8293, 8297, 8317, 8329, 8369, 8377, 8389, 8423, 8431, 8447, 8467, 8527, 8539, 8543, 8581, 8599, 8609, 8629, 8647, 8669, 8681, 8693, 8699, 8719, 8741, 8753, 8761, 8783, 8807, 8821, 8839, 8849, 8863, 8867, 8893, 8933, 8941, 8951, 8971, 9001, 9013, 9043, 9049, 9067, 9109, 9137, 9161, 9187, 9203, 9209, 9227, 9241, 9257, 9283, 9293, 9323, 9343, 9349, 9377, 9403, 9421, 9433, 9439, 9463, 9467, 9479, 9497, 9521, 9539, 9551, 9623, 9631, 9649, 9661, 9679, 9697, 9721, 9743, 9749, 9769, 9791, 9817, 9833, 9839, 9859, 9887, 9907, 9931, 9949, 9973, 10009, 10039, 10069, 10079, 10093, 10103, 10111, 10141, 10163, 10169, 10181.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 2797476 values, from 3 to 99999971).

n\r 0  1 
202797476 2 
3114411461356329 3 
40139892801398548 4 
50664969610964779622741921 5 
6014411460101356329 6 
71498724433369488548437813497432441589 7 
80698320069903907006080699509 8 
9047988045193714802104520400481056452352 9 
10066496907796220006109640741921 10 
110276374250004287722262093314351251732295347269610305783284460

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.