Noncototient
In number theory, a noncototient is a positive integer Template:Mvar that cannot be expressed as the difference between a positive integer Template:Mvar and the number of coprime integers below it. That is, Template:Math, where Template:Mvar stands for Euler's totient function, has no solution for Template:Mvar. The cototient of Template:Mvar is defined as Template:Math, 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 Template:Mvar can be represented as a sum of two distinct primes Template:Mvar and Template:Mvar, then
<math display=block>\begin{align}
pq - \varphi(pq) &= pq - (p-1)(q-1) \\ &= p + q - 1 \\ &= n - 1.
\end{align}</math>
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 Template:Math, Template:Math, and Template:Math.
For even numbers, it can be shown <math display=block>\begin{align}
2pq - \varphi(2pq) &= 2pq - (p-1)(q-1) \\ &= pq + p + q - 1 \\ &= (p+1)(q+1) - 2
\end{align}</math>
Thus, all even numbers Template:Mvar such that Template:Math can be written as Template:Math with Template:Mvar 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, ... (sequence A005278 in the OEIS)
The cototient of Template:Mvar 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, ... (sequence A051953 in the OEIS)
Least Template:Mvar such that the cototient of Template:Mvar is Template:Mvar are (start with Template:Math, 0 if no such Template:Mvar 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, ... (sequence A063507 in the OEIS)
Greatest Template:Mvar such that the cototient of Template:Mvar is Template:Mvar are (start with Template:Math, 0 if no such Template:Mvar 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, ... (sequence A063748 in the OEIS)
Number of Template:Mvars such that Template:Math is Template:Mvar are (start with Template:Math)
- 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, ... (sequence A063740 in the OEIS)
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 <math> 2^k \cdot 509203</math> is an example (See Riesel number). Since then other infinite families, of roughly the same form, have been given by Flammenkamp and Luca (2000).
| Template:Mvar | Numbers Template:Mvar such that Template:Math |
|---|---|
| 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 |