The Pierpont primes up to 1015 :
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.
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