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 |
2 | 0 | 2797476 | 2 |
3 | 1 | 1441146 | 1356329 | 3 |
4 | 0 | 1398928 | 0 | 1398548 | 4 |
5 | 0 | 664969 | 610964 | 779622 | 741921 | 5 |
6 | 0 | 1441146 | 0 | 1 | 0 | 1356329 | 6 |
7 | 1 | 498724 | 433369 | 488548 | 437813 | 497432 | 441589 | 7 |
8 | 0 | 698320 | 0 | 699039 | 0 | 700608 | 0 | 699509 | 8 |
9 | 0 | 479880 | 451937 | 1 | 480210 | 452040 | 0 | 481056 | 452352 | 9 |
10 | 0 | 664969 | 0 | 779622 | 0 | 0 | 0 | 610964 | 0 | 741921 | 10 |
11 | 0 | 276374 | 250004 | 287722 | 262093 | 314351 | 251732 | 295347 | 269610 | 305783 | 284460 |
A pictorial representation of the table above
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.