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Proth numbers
A number of the form k ⋅ 2n + 1, with k odd, n > 0, and 2n > k. more

The first 600 Proth numbers :
3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 289, 321, 353, 385, 417, 449, 481, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801, 833, 865, 897, 929, 961, 993, 1025, 1089, 1153, 1217, 1281, 1345, 1409, 1473, 1537, 1601, 1665, 1729, 1793, 1857, 1921, 1985, 2049, 2113, 2177, 2241, 2305, 2369, 2433, 2497, 2561, 2625, 2689, 2753, 2817, 2881, 2945, 3009, 3073, 3137, 3201, 3265, 3329, 3393, 3457, 3521, 3585, 3649, 3713, 3777, 3841, 3905, 3969, 4033, 4097, 4225, 4353, 4481, 4609, 4737, 4865, 4993, 5121, 5249, 5377, 5505, 5633, 5761, 5889, 6017, 6145, 6273, 6401, 6529, 6657, 6785, 6913, 7041, 7169, 7297, 7425, 7553, 7681, 7809, 7937, 8065, 8193, 8321, 8449, 8577, 8705, 8833, 8961, 9089, 9217, 9345, 9473, 9601, 9729, 9857, 9985, 10113, 10241, 10369, 10497, 10625, 10753, 10881, 11009, 11137, 11265, 11393, 11521, 11649, 11777, 11905, 12033, 12161, 12289, 12417, 12545, 12673, 12801, 12929, 13057, 13185, 13313, 13441, 13569, 13697, 13825, 13953, 14081, 14209, 14337, 14465, 14593, 14721, 14849, 14977, 15105, 15233, 15361, 15489, 15617, 15745, 15873, 16001, 16129, 16257, 16385, 16641, 16897, 17153, 17409, 17665, 17921, 18177, 18433, 18689, 18945, 19201, 19457, 19713, 19969, 20225, 20481, 20737, 20993, 21249, 21505, 21761, 22017, 22273, 22529, 22785, 23041, 23297, 23553, 23809, 24065, 24321, 24577, 24833, 25089, 25345, 25601, 25857, 26113, 26369, 26625, 26881, 27137, 27393, 27649, 27905, 28161, 28417, 28673, 28929, 29185, 29441, 29697, 29953, 30209, 30465, 30721, 30977, 31233, 31489, 31745, 32001, 32257, 32513, 32769, 33025, 33281, 33537, 33793, 34049, 34305, 34561, 34817, 35073, 35329, 35585, 35841, 36097, 36353, 36609, 36865, 37121, 37377, 37633, 37889, 38145, 38401, 38657, 38913, 39169, 39425, 39681, 39937, 40193, 40449, 40705, 40961, 41217, 41473, 41729, 41985, 42241, 42497, 42753, 43009, 43265, 43521, 43777, 44033, 44289, 44545, 44801, 45057, 45313, 45569, 45825, 46081, 46337, 46593, 46849, 47105, 47361, 47617, 47873, 48129, 48385, 48641, 48897, 49153, 49409, 49665, 49921, 50177, 50433, 50689, 50945, 51201, 51457, 51713, 51969, 52225, 52481, 52737, 52993, 53249, 53505, 53761, 54017, 54273, 54529, 54785, 55041, 55297, 55553, 55809, 56065, 56321, 56577, 56833, 57089, 57345, 57601, 57857, 58113, 58369, 58625, 58881, 59137, 59393, 59649, 59905, 60161, 60417, 60673, 60929, 61185, 61441, 61697, 61953, 62209, 62465, 62721, 62977, 63233, 63489, 63745, 64001, 64257, 64513, 64769, 65025, 65281, 65537, 66049, 66561, 67073, 67585, 68097, 68609, 69121, 69633, 70145, 70657, 71169, 71681, 72193, 72705, 73217, 73729, 74241, 74753, 75265, 75777, 76289, 76801, 77313, 77825, 78337, 78849, 79361, 79873, 80385, 80897, 81409, 81921, 82433, 82945, 83457, 83969, 84481, 84993, 85505, 86017, 86529, 87041, 87553, 88065, 88577, 89089, 89601, 90113, 90625, 91137, 91649, 92161, 92673, 93185, 93697, 94209, 94721, 95233, 95745, 96257, 96769, 97281, 97793, 98305, 98817, 99329, 99841, 100353, 100865, 101377, 101889, 102401, 102913, 103425, 103937, 104449, 104961, 105473, 105985, 106497, 107009, 107521, 108033, 108545, 109057, 109569, 110081, 110593, 111105, 111617, 112129, 112641, 113153, 113665, 114177, 114689, 115201, 115713, 116225, 116737, 117249, 117761, 118273, 118785, 119297, 119809, 120321, 120833, 121345, 121857, 122369, 122881, 123393, 123905, 124417, 124929, 125441, 125953, 126465, 126977, 127489, 128001, 128513, 129025, 129537, 130049, 130561, 131073, 131585, 132097, 132609, 133121, 133633, 134145, 134657, 135169, 135681, 136193, 136705, 137217, 137729, 138241, 138753, 139265, 139777, 140289, 140801, 141313, 141825, 142337, 142849, 143361, 143873, 144385, 144897, 145409, 145921, 146433, 146945, 147457, 147969, 148481, 148993, 149505, 150017, 150529, 151041, 151553, 152065, 152577, 153089, 153601, 154113, 154625, 155137, 155649, 156161, 156673, 157185, 157697, 158209, 158721, 159233, 159745, 160257, 160769, 161281, 161793, 162305, 162817, 163329, 163841, 164353, 164865, 165377, 165889, 166401, 166913, 167425, 167937, 168449, 168961, 169473, 169985, 170497, 171009, 171521, 172033, 172545, 173057, 173569, 174081, 174593, 175105, 175617, 176129, 176641.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1477961 values, from 3 to 999999668225).

n\r 0  1 
201477961 2 
3492654492653492654 3 
40147796001 4 
5295593295591295592295593295592 5 
6049265304926540492654 6 
7211131211137211137211138211137211138211143 7 
801477958010200 8 
9164218164217164218164219164218164218164217164218164218 9 
1002955910295593029559302955920295592 10 
11134360134359134360134361134360134361134360134361134360134360134359

A pictorial representation of the table above
motab
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.