A prime pn such that pn2 > pn-i ⋅ pn+i for every 0 < i < n. more
The first 600 good primes :
5,
11,
17,
29,
37,
41,
53,
59,
67,
71,
97,
101,
127,
149,
179,
191,
223,
227,
251,
257,
269,
307,
311,
331,
347,
419,
431,
541,
557,
563,
569,
587,
593,
599,
641,
727,
733,
739,
809,
821,
853,
929,
937,
967,
1009,
1031,
1087,
1151,
1213,
1277,
1367,
1399,
1423,
1427,
1543,
1597,
1847,
1861,
1867,
1871,
1973,
1987,
1993,
1997,
2063,
2203,
2237,
2267,
2333,
2339,
2521,
2531,
2539,
2609,
2647,
2657,
2677,
2683,
2687,
2999,
3163,
3167,
3251,
3299,
3433,
3449,
3457,
3461,
3511,
3527,
3821,
3847,
3907,
3989,
4001,
4201,
4211,
4217,
4229,
4441,
4447,
4481,
4507,
4621,
4637,
4783,
4861,
4871,
4903,
4931,
4967,
5381,
5387,
5407,
5413,
5623,
5639,
5647,
5651,
6007,
6029,
6037,
6043,
6067,
6197,
6257,
6263,
6521,
6547,
6551,
6563,
6637,
6653,
6659,
6689,
6761,
6779,
6823,
6947,
7411,
7451,
7457,
7477,
7517,
7523,
7537,
8039,
8053,
8081,
8087,
8161,
8167,
8209,
8219,
8501,
8513,
8521,
8597,
8623,
8663,
8677,
8689,
8999,
9103,
9127,
9133,
9151,
9277,
9311,
9319,
9337,
9371,
9391,
9397,
9413,
9613,
10007,
10037,
10061,
10067,
10091,
10133,
10589,
10597,
10831,
10847,
10853,
11047,
11057,
11239,
11657,
11677,
11699,
11777,
11807,
11887,
11897,
12197,
12227,
12239,
12373,
12377,
12473,
12487,
12889,
13669,
13679,
13687,
14243,
14293,
14321,
14369,
14387,
14401,
14407,
14519,
14533,
14699,
14713,
15013,
15053,
15073,
15217,
15227,
15259,
15269,
16411,
16417,
16547,
16811,
16823,
16829,
16871,
16879,
16979,
17291,
17317,
17377,
17383,
17783,
17827,
17837,
17881,
17903,
17909,
18899,
18911,
19001,
19139,
19207,
19373,
19379,
19417,
19421,
19889,
19913,
19961,
20681,
20707,
20717,
20743,
20849,
20873,
20897,
20981,
21139,
21313,
21377,
21467,
21481,
21487,
21491,
21557,
22531,
22541,
22613,
22619,
22637,
22691,
22697,
22961,
23003,
23497,
23531,
23537,
23549,
23557,
23741,
24763,
24841,
24907,
24917,
25087,
25111,
25117,
25147,
25301,
25523,
25537,
25561,
25577,
25913,
26669,
26681,
27397,
27407,
27427,
27611,
27673,
27689,
27733,
27737,
28277,
28387,
28403,
28493,
28537,
28571,
28591,
28597,
29833,
29983,
30011,
30059,
30089,
30491,
30631,
30637,
30671,
30757,
30803,
31121,
31139,
31147,
31957,
32027,
32051,
32057,
32297,
32969,
33287,
33311,
33329,
34123,
35729,
35747,
35797,
35801,
35831,
35951,
35963,
36433,
36451,
36467,
36523,
36527,
36671,
38113,
38149,
38167,
38177,
38543,
38557,
38593,
38651,
38669,
39079,
39089,
40423,
40427,
40459,
40471,
40483,
40693,
40697,
40739,
40751,
40759,
40801,
40813,
40819,
41011,
41039,
41113,
41131,
41141,
41177,
41507,
41513,
41579,
41593,
41603,
41843,
41849,
41879,
41887,
41941,
41947,
43573,
43577,
43889,
43933,
43943,
43961,
44449,
44483,
44491,
44497,
45817,
45943,
46021,
46049,
46091,
46133,
46141,
46399,
46439,
46993,
47041,
47051,
47087,
47111,
47119,
47269,
47279,
47287,
47293,
47491,
47497,
48299,
48311,
48337,
48397,
48407,
48463,
48473,
48479,
48731,
48751,
48757,
48973,
48989,
49103,
50741,
50767,
50821,
50833,
51109,
51131,
51193,
51197,
51329,
51341,
51407,
51413,
51419,
52501,
52511,
52529,
52541,
53591,
54251,
54269,
54311,
54319,
54347,
54361,
54367,
54401,
55579,
55603,
55619,
55661,
55763,
55787,
55793,
56359,
56369,
56377,
56431,
56437,
56467,
56473,
58889,
58897,
60589,
60601,
60607,
61211,
61253,
61283,
61331,
61441,
61463,
61469,
62459,
62467,
63241,
63277,
63299,
63311,
64553,
64567,
64577,
64601,
64849,
64871,
64877,
64997,
65027,
65089,
65099,
65353,
65407,
65519,
65537,
66293,
66337,
66343,
66359,
66449,
66457,
66491,
66697,
66841,
66851,
66919,
67033,
67121,
69653,
69677,
69737,
69761,
69809,
69821,
69827,
69991,
69997,
70001,
70099,
70111,
70117,
70823,
70841,
70913,
71143,
71233,
73277,
73291,
73303,
73327,
73351,
73361,
73517,
73523,
73547,
73939,
73999,
74017,
74047,
74131,
74143,
74159,
74687,
74699,
74707,
75133,
75149,
75161,
75167,
76463,
77137,
77237,
77261,
77417,
77471,
77477,
79031,
79087,
79103,
79133,
79147,
79273,
79531,
79549,
79601,
79757,
79769,
79801,
79811,
80141,
80147,
80447,
80471,
80599,
80651,
80669,
80909,
84047,
84053.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 68658 values, from 5 to 199968539).
n\r | 0 | 1 |
2 | 0 | 68658 | 2 |
3 | 0 | 33212 | 35446 | 3 |
4 | 0 | 34006 | 0 | 34652 | 4 |
5 | 1 | 17976 | 19373 | 15214 | 16094 | 5 |
6 | 0 | 33212 | 0 | 0 | 0 | 35446 | 6 |
7 | 0 | 10340 | 12033 | 11058 | 12318 | 10498 | 12411 | 7 |
8 | 0 | 17033 | 0 | 17340 | 0 | 16973 | 0 | 17312 | 8 |
9 | 0 | 11073 | 11811 | 0 | 11184 | 11878 | 0 | 10955 | 11757 | 9 |
10 | 0 | 17976 | 0 | 15214 | 0 | 1 | 0 | 19373 | 0 | 16094 | 10 |
11 | 1 | 7090 | 7476 | 6606 | 7162 | 6037 | 7593 | 6604 | 7147 | 6266 | 6676 |
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.