A prime of the form 1 + 2u ⋅ 3v. more
The Pierpont primes up to 10
15 :
2,
3,
5,
7,
13,
17,
19,
37,
73,
97,
109,
163,
193,
257,
433,
487,
577,
769,
1153,
1297,
1459,
2593,
2917,
3457,
3889,
10369,
12289,
17497,
18433,
39367,
52489,
65537,
139969,
147457,
209953,
331777,
472393,
629857,
746497,
786433,
839809,
995329,
1179649,
1492993,
1769473,
1990657,
2654209,
5038849,
5308417,
8503057,
11337409,
14155777,
19131877,
28311553,
57395629,
63700993,
71663617,
86093443,
102036673,
113246209,
120932353,
169869313,
258280327,
483729409,
725594113,
1088391169,
1811939329,
2717908993,
3221225473,
3439853569,
4076863489,
6879707137,
10871635969,
11609505793,
18345885697,
29386561537,
69657034753,
77309411329,
123834728449,
206158430209,
251048476873,
347892350977,
880602513409,
1253826625537,
1410554953729,
1761205026817,
2348273369089,
2380311484417,
2783138807809,
4518872583697,
5566277615617,
6347497291777,
6597069766657,
14281868906497,
17832200896513,
22568879259649,
25048249270273,
29686813949953,
33853318889473,
39582418599937,
44530220924929,
56358560858113,
79164837199873,
84537841287169,
85691213438977,
90275517038593,
126806761930753,
142657607172097,
150289495621633,
270826551115777,
411782264189299,
457019805007873,
474989023199233,
578415690713089.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 114 values, from 2 to 578415690713089).
n\r | 0 | 1 |
2 | 1 | 113 | 2 |
3 | 1 | 108 | 5 | 3 |
4 | 0 | 103 | 1 | 10 | 4 |
5 | 1 | 0 | 41 | 37 | 35 | 5 |
6 | 0 | 108 | 1 | 1 | 0 | 4 | 6 |
7 | 1 | 0 | 18 | 23 | 25 | 26 | 21 | 7 |
8 | 0 | 96 | 1 | 6 | 0 | 7 | 0 | 4 | 8 |
9 | 0 | 98 | 1 | 1 | 7 | 2 | 0 | 3 | 2 | 9 |
10 | 0 | 0 | 1 | 37 | 0 | 1 | 0 | 40 | 0 | 35 | 10 |
11 | 0 | 0 | 12 | 14 | 11 | 8 | 16 | 15 | 14 | 16 | 8 |
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.