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 |
2 | 0 | 1477961 | 2 |
3 | 492654 | 492653 | 492654 | 3 |
4 | 0 | 1477960 | 0 | 1 | 4 |
5 | 295593 | 295591 | 295592 | 295593 | 295592 | 5 |
6 | 0 | 492653 | 0 | 492654 | 0 | 492654 | 6 |
7 | 211131 | 211137 | 211137 | 211138 | 211137 | 211138 | 211143 | 7 |
8 | 0 | 1477958 | 0 | 1 | 0 | 2 | 0 | 0 | 8 |
9 | 164218 | 164217 | 164218 | 164219 | 164218 | 164218 | 164217 | 164218 | 164218 | 9 |
10 | 0 | 295591 | 0 | 295593 | 0 | 295593 | 0 | 295592 | 0 | 295592 | 10 |
11 | 134360 | 134359 | 134360 | 134361 | 134360 | 134361 | 134360 | 134361 | 134360 | 134360 | 134359 |
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.