A prime which is the average of the two surrounding primes. more
The first 600 balanced primes :
5,
53,
157,
173,
211,
257,
263,
373,
563,
593,
607,
653,
733,
947,
977,
1103,
1123,
1187,
1223,
1367,
1511,
1747,
1753,
1907,
2287,
2417,
2677,
2903,
2963,
3307,
3313,
3637,
3733,
4013,
4409,
4457,
4597,
4657,
4691,
4993,
5107,
5113,
5303,
5387,
5393,
5563,
5807,
6073,
6263,
6317,
6323,
6367,
6373,
6863,
6977,
7523,
7583,
7823,
7841,
8117,
8713,
8747,
9397,
9473,
9871,
10253,
10607,
10657,
10853,
11299,
11411,
11497,
11731,
11807,
11903,
11933,
12497,
12547,
12583,
12647,
12653,
12841,
12973,
13043,
13177,
13457,
13463,
14543,
14747,
14753,
15161,
15193,
15313,
15467,
15767,
15797,
15803,
15907,
15913,
16097,
16223,
16427,
16487,
16567,
16619,
16787,
16937,
16987,
17047,
17327,
17431,
17477,
17483,
17851,
18217,
18223,
18341,
18433,
18731,
19463,
19477,
19483,
19577,
19603,
19739,
19949,
20107,
20123,
20161,
20201,
20347,
20521,
20731,
21163,
21661,
21911,
21997,
22051,
22073,
22259,
22447,
23327,
23333,
23767,
23801,
23893,
24077,
24097,
24413,
24677,
25391,
25463,
25621,
25673,
26041,
26177,
26183,
26393,
26693,
26723,
26987,
27067,
27773,
28069,
28597,
28927,
29173,
29269,
29333,
29599,
29873,
30059,
30097,
30103,
30313,
30449,
30637,
30643,
30977,
31051,
31327,
32083,
32479,
32497,
32993,
33113,
33857,
34171,
34267,
34543,
34631,
34673,
34877,
35129,
35317,
35753,
35851,
35911,
36067,
36229,
36313,
36677,
36821,
37957,
38183,
38867,
39113,
39863,
40063,
40093,
40111,
40283,
40471,
40493,
40933,
40961,
41017,
41263,
41479,
41513,
41603,
41947,
42187,
42209,
42397,
42533,
42683,
43189,
44041,
44519,
44543,
44741,
45433,
45497,
45757,
46147,
46889,
47303,
47507,
47527,
47869,
48023,
48259,
48533,
48661,
49069,
49633,
50153,
50417,
50441,
51413,
51461,
51551,
52529,
52727,
52957,
53617,
53623,
53939,
54151,
54443,
54623,
54673,
54773,
54851,
55207,
55793,
55843,
56003,
56087,
56093,
56149,
56437,
56773,
56957,
57119,
57853,
58067,
58129,
58237,
58543,
58573,
58693,
58979,
59113,
59393,
60083,
60133,
60337,
60509,
60617,
61553,
61637,
61723,
62207,
62213,
62597,
62633,
63073,
63493,
63527,
63559,
63697,
63703,
63907,
64013,
64627,
65413,
65557,
65707,
65957,
66431,
66809,
66883,
67169,
67511,
67619,
67783,
68099,
68483,
68699,
68743,
69067,
69623,
71011,
71347,
71353,
71843,
71993,
72031,
72481,
72733,
73127,
73477,
73523,
74317,
74471,
74489,
74567,
74897,
75181,
75217,
75527,
75533,
75577,
75641,
75787,
76487,
76561,
76579,
76673,
76913,
77023,
77081,
77557,
77563,
77951,
78173,
78259,
78577,
78797,
78803,
79187,
79411,
79823,
79867,
79973,
80251,
80329,
80917,
80923,
81001,
81077,
81439,
81689,
81749,
81937,
82463,
82493,
82787,
82793,
83003,
83437,
83443,
84053,
84137,
84437,
84443,
85049,
85121,
85837,
86137,
86263,
86981,
87427,
87517,
87547,
88867,
89003,
89107,
89113,
89387,
89393,
89527,
89983,
90511,
90703,
91243,
91297,
91303,
91387,
92003,
92227,
92363,
92581,
93053,
93083,
93377,
94427,
94433,
95267,
95273,
95707,
95917,
95923,
96001,
96377,
96431,
96763,
97441,
97967,
98473,
98597,
99823,
100937,
101273,
101287,
101483,
103177,
103567,
103669,
104003,
104021,
104053,
104113,
104161,
104207,
104543,
104717,
104723,
104953,
105367,
105373,
105607,
105613,
105977,
106297,
106681,
107581,
107693,
108217,
108439,
108637,
108643,
108739,
108923,
109897,
110459,
110603,
110681,
110813,
110933,
111509,
111521,
111611,
111773,
111863,
112181,
112297,
112583,
112939,
113117,
113921,
113963,
114001,
114083,
114473,
114901,
115337,
115757,
115763,
116107,
116719,
116797,
117437,
117757,
118831,
119033,
119267,
119557,
119563,
119759,
120097,
120557,
120563,
120817,
120823,
121007,
121013,
121327,
121421,
121447,
121553,
122099,
122273,
122579,
122833,
123433,
123887,
124459,
125107,
125441,
125539,
125737,
126211,
126317,
126839,
127487,
127643,
127663,
127733,
127843,
128153,
128449,
128467,
128483,
129049,
129209,
129763,
130343,
130483,
130687,
130693,
130829,
131041,
131933,
132511,
132911,
132929,
133103,
133571,
133717,
134053,
134213,
134263,
134333,
134639,
135089,
135301,
135607,
135893,
136733,
137131,
137321,
137393,
137849,
138107,
139079,
139703,
139753,
140351,
140527,
140983,
141067,
141073,
141263,
142019,
142111,
142217,
142369,
142427,
142553,
143159,
143483,
143699,
144247,
144253,
144439,
144583,
144611,
144757,
144967,
145037,
145121,
145213,
145589,
145661,
145879,
146197,
146701.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 10801518 values, from 5 to 9999997543).
n\r | 0 | 1 |
2 | 0 | 10801518 | 2 |
3 | 0 | 5399872 | 5401646 | 3 |
4 | 0 | 5400730 | 0 | 5400788 | 4 |
5 | 1 | 2429250 | 2972579 | 2971162 | 2428526 | 5 |
6 | 0 | 5399872 | 0 | 0 | 0 | 5401646 | 6 |
7 | 0 | 1515497 | 1790713 | 2096795 | 2095009 | 1787089 | 1516415 | 7 |
8 | 0 | 2702288 | 0 | 2699637 | 0 | 2698442 | 0 | 2701151 | 8 |
9 | 0 | 1797037 | 1801879 | 0 | 1800396 | 1800096 | 0 | 1802439 | 1799671 | 9 |
10 | 0 | 2429250 | 0 | 2971162 | 0 | 1 | 0 | 2972579 | 0 | 2428526 | 10 |
11 | 0 | 1025872 | 1225106 | 1191260 | 1165908 | 795210 | 795503 | 1164870 | 1189058 | 1224522 | 1024209 |
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.