In number theory, a noncototient is a positive integer that cannot be expressed as the difference between a positive integer and the number of coprime integers below it. That is,, where stands for Euler's totient function, has no solution for . The cototient of is defined as, so a noncototient is a number that is never a cototient.
It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number can be represented as a sum of two distinct primes and, then
It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations,, and .
For even numbers, it can be shown
Thus, all even numbers such that can be written as with primes are cototients.
The first few noncototients are
10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, 206, 218, 222, 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, ...
The cototient of are
0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, 1, 8, 7, 8, 1, 12, 1, 12, 9, 12, 1, 16, 5, 14, 9, 16, 1, 22, 1, 16, 13, 18, 11, 24, 1, 20, 15, 24, 1, 30, 1, 24, 21, 24, 1, 32, 7, 30, 19, 28, 1, 36, 15, 32, 21, 30, 1, 44, 1, 32, 27, 32, 17, 46, 1, 36, 25, 46, 1, 48, ...
Least such that the cototient of is are (start with, 0 if no such exists)
1, 2, 4, 9, 6, 25, 10, 15, 12, 21, 0, 35, 18, 33, 26, 39, 24, 65, 34, 51, 38, 45, 30, 95, 36, 69, 0, 63, 52, 161, 42, 87, 48, 93, 0, 75, 54, 217, 74, 99, 76, 185, 82, 123, 60, 117, 66, 215, 72, 141, 0, ...
Greatest such that the cototient of is are (start with, 0 if no such exists)
1, ∞, 4, 9, 8, 25, 10, 49, 16, 27, 0, 121, 22, 169, 26, 55, 32, 289, 34, 361, 38, 85, 30, 529, 46, 133, 0, 187, 52, 841, 58, 961, 64, 253, 0, 323, 68, 1369, 74, 391, 76, 1681, 82, 1849, 86, 493, 70, 2209, 94, 589, 0, ...
Number of s such that is are (start with)
1, ∞, 1, 1, 2, 1, 1, 2, 3, 2, 0, 2, 3, 2, 1, 2, 3, 3, 1, 3, 1, 3, 1, 4, 4, 3, 0, 4, 1, 4, 3, 3, 4, 3, 0, 5, 2, 2, 1, 4, 1, 5, 1, 4, 2, 4, 2, 6, 5, 5, 0, 3, 0, 6, 2, 4, 2, 5, 0, 7, 4, 3, 1, 8, 4, 6, 1, 3, 1, 5, 2, 7, 3, ...
Erdős (1913–1996) and Sierpinski (1882–1969) asked whether there exist infinitely many noncototients. This was finally answered in the affirmative by Browkin and Schinzel (1995), who showed every member of the infinite family
2k ⋅ 509203
1 | all primes | |
---|---|---|
2 | 4 | |
3 | 9 | |
4 | 6, 8 | |
5 | 25 | |
6 | 10 | |
7 | 15, 49 | |
8 | 12, 14, 16 | |
9 | 21, 27 | |
10 | ||
11 | 35, 121 | |
12 | 18, 20, 22 | |
13 | 33, 169 | |
14 | 26 | |
15 | 39, 55 | |
16 | 24, 28, 32 | |
17 | 65, 77, 289 | |
18 | 34 | |
19 | 51, 91, 361 | |
20 | 38 | |
21 | 45, 57, 85 | |
22 | 30 | |
23 | 95, 119, 143, 529 | |
24 | 36, 40, 44, 46 | |
25 | 69, 125, 133 | |
26 | ||
27 | 63, 81, 115, 187 | |
28 | 52 | |
29 | 161, 209, 221, 841 | |
30 | 42, 50, 58 | |
31 | 87, 247, 961 | |
32 | 48, 56, 62, 64 | |
33 | 93, 145, 253 | |
34 | ||
35 | 75, 155, 203, 299, 323 | |
36 | 54, 68 | |
37 | 217, 1369 | |
38 | 74 | |
39 | 99, 111, 319, 391 | |
40 | 76 | |
41 | 185, 341, 377, 437, 1681 | |
42 | 82 | |
43 | 123, 259, 403, 1849 | |
44 | 60, 86 | |
45 | 117, 129, 205, 493 | |
46 | 66, 70 | |
47 | 215, 287, 407, 527, 551, 2209 | |
48 | 72, 80, 88, 92, 94 | |
49 | 141, 301, 343, 481, 589 | |
50 | ||
51 | 235, 451, 667 | |
52 | ||
53 | 329, 473, 533, 629, 713, 2809 | |
54 | 78, 106 | |
55 | 159, 175, 559, 703 | |
56 | 98, 104 | |
57 | 105, 153, 265, 517, 697 | |
58 | ||
59 | 371, 611, 731, 779, 851, 899, 3481 | |
60 | 84, 100, 116, 118 | |
61 | 177, 817, 3721 | |
62 | 122 | |
63 | 135, 147, 171, 183, 295, 583, 799, 943 | |
64 | 96, 112, 124, 128 | |
65 | 305, 413, 689, 893, 989, 1073 | |
66 | 90 | |
67 | 427, 1147, 4489 | |
68 | 134 | |
69 | 201, 649, 901, 1081, 1189 | |
70 | 102, 110 | |
71 | 335, 671, 767, 1007, 1247, 1271, 5041 | |
72 | 108, 136, 142 | |
73 | 213, 469, 793, 1333, 5329 | |
74 | 146 | |
75 | 207, 219, 275, 355, 1003, 1219, 1363 | |
76 | 148 | |
77 | 245, 365, 497, 737, 1037, 1121, 1457, 1517 | |
78 | 114 | |
79 | 511, 871, 1159, 1591, 6241 | |
80 | 152, 158 | |
81 | 189, 237, 243, 781, 1357, 1537 | |
82 | 130 | |
83 | 395, 803, 923, 1139, 1403, 1643, 1739, 1763, 6889 | |
84 | 164, 166 | |
85 | 165, 249, 325, 553, 949, 1273 | |
86 | ||
87 | 415, 1207, 1711, 1927 | |
88 | 120, 172 | |
89 | 581, 869, 1241, 1349, 1541, 1769, 1829, 1961, 2021, 7921 | |
90 | 126, 178 | |
91 | 267, 1027, 1387, 1891 | |
92 | 132, 140 | |
93 | 261, 445, 913, 1633, 2173 | |
94 | 138, 154 | |
95 | 623, 1079, 1343, 1679, 1943, 2183, 2279 | |
96 | 144, 160, 176, 184, 188 | |
97 | 1501, 2077, 2257, 9409 | |
98 | 194 | |
99 | 195, 279, 291, 979, 1411, 2059, 2419, 2491 | |
100 | ||
101 | 485, 1157, 1577, 1817, 2117, 2201, 2501, 2537, 10201 | |
102 | 202 | |
103 | 303, 679, 2263, 2479, 2623, 10609 | |
104 | 206 | |
105 | 225, 309, 425, 505, 1513, 1909, 2773 | |
106 | 170 | |
107 | 515, 707, 1067, 1691, 2291, 2627, 2747, 2867, 11449 | |
108 | 156, 162, 212, 214 | |
109 | 321, 721, 1261, 2449, 2701, 2881, 11881 | |
110 | 150, 182, 218 | |
111 | 231, 327, 535, 1111, 2047, 2407, 2911, 3127 | |
112 | 196, 208 | |
113 | 545, 749, 1133, 1313, 1649, 2573, 2993, 3053, 3149, 3233, 12769 | |
114 | 226 | |
115 | 339, 475, 763, 1339, 1843, 2923, 3139 | |
116 | ||
117 | 297, 333, 565, 1177, 1717, 2581, 3337 | |
118 | 174, 190 | |
119 | 539, 791, 1199, 1391, 1751, 1919, 2231, 2759, 3071, 3239, 3431, 3551, 3599 | |
120 | 168, 200, 232, 236 | |
121 | 1331, 1417, 1957, 3397 | |
122 | ||
123 | 1243, 1819, 2323, 3403, 3763 | |
124 | 244 | |
125 | 625, 1469, 1853, 2033, 2369, 2813, 3293, 3569, 3713, 3869, 3953 | |
126 | 186 | |
127 | 255, 2071, 3007, 4087, 16129 | |
128 | 192, 224, 248, 254, 256 | |
129 | 273, 369, 381, 1921, 2461, 2929, 3649, 3901, 4189 | |
130 | ||
131 | 635, 2147, 2507, 2987, 3131, 3827, 4187, 4307, 4331, 17161 | |
132 | 180, 242, 262 | |
133 | 393, 637, 889, 3193, 3589, 4453 | |
134 | ||
135 | 351, 387, 575, 655, 2599, 3103, 4183, 4399 | |
136 | 268 | |
137 | 917, 1397, 3161, 3317, 3737, 3977, 4661, 4757, 18769 | |
138 | 198, 274 | |
139 | 411, 1651, 3379, 3811, 4171, 4819, 4891, 19321 | |
140 | 204, 220, 278 | |
141 | 285, 417, 685, 1441, 3277, 4141, 4717, 4897 | |
142 | 230, 238 | |
143 | 363, 695, 959, 1703, 2159, 3503, 3959, 4223, 4343, 4559, 5063, 5183 | |
144 | 216, 272, 284 |