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In number theory, a nontotient is a positive integer n which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(x) = n has no solution x. In other words, n is a nontotient if there is no integer x that has exactly n coprimes below it. All odd numbers are nontotients, except 1, since it has the solutions x = 1 and x = 2. The first few even nontotients are

12, 22, 2X, 32, 42, 52, 58, 62, 64, 72, 76, 7X, 82, 96, 9X, X2, X4, E2, EX, 102, 108, 10X, 112, 122, 126, 132, 136, 138, 142, 14X, 152, 15X, 162, 172, 176, 178, 182, 184, 186, 188, 192, 196, 1X2, 1XX, 1E2, 1E8, 1EX, 202, 20X, 212, 214, 218, 222, 226, 22X, 232, 23X, 242, 244, 252, 256, 262, 264, 26X, 272, 274, 282, 286, 28X, 292, 296, 298, 29X, 2X2, 2X4, 2X6, 2E2, 2E6, 2E8, 302, 304, 312, 316, 31X, 322, 332, 334, 336, 342, 344, 348, 352, 354, 366, 36X, 370, 372, 37X, 382, 384, 386, 38X, 392, 398, 39X, 3X2, 3X6, 3E2, 3E8, 3EX, 402, 406, 412, 416, 418, 422, 424, 428, 42X, 432, 43X, 442, 444, 44X, 452, 458, 462, 466, 472, 476, 478, 47X, 482, 486, 488, 492, 49X, 4X2, 4X6, 4XX, 4E2, 4E6, 502, 504, 508, 50X, 512, 518, 522, 524, 528, 52X, 532, 536, 53X, 542, 546, 54X, 552, 558, 55X, 562, 566, 56X, 570, 572, 57X, 582, 588, 592, 596, 5X2, 5X4, 5X6, 5XX, 5E2, 602, 604, 606, 608, 60X, 612, 622, 624, 626, 62X, 632, 638, 642, 644, 64X, 652, 65X, 662, 666, 668, 670, 672, 676, 67X, 682, 684, 688, 692, 696, 6X2, 6X4, 6X8, 6XX, 6E2, 6E6, 6EX, 702, 706, 712, 716, 718, 722, 724, 72X, 732, 738, 73X, 742, 748, 74X, 752, 756, 758, 75X, 762, 764, 772, 776, 778, 782, 786, 78X, 792, 79X, 7X2, 7X4, 7X6, 7XX, 7E2, 7E6, 7E8, 802, 804, 806, 810, 812, 81X, 822, 826, 832, 836, 83X, 842, 844, 846, 848, 852, 856, 858, 862, 86X, 872, 876, 878, 87X, 882, 884, 886, 888, 892, 896, 898, 89X, 8X2, 8E2, 8EX, 902, 910, 912, 916, 922, 924, 92X, 932, 934, 936, 938, 93X, 942, 944, 946, 94X, 952, 956, 962, 966, 96X, 972, 976, 978, 97X, 982, 984, 98X, 992, 996, 998, 99X, 9X2, 9X4, 9E2, 9E6, X02, X04, X08, X12, X18, X1X, X22, X24, X2X, X32, X42, X46, X48, X52, X54, X56, X58, X62, X64, X66, X70, X72, X78, X7X, X82, X8X, X92, X96, X98, XX2, XX4, XE2, XE4, XE8, E02, E04, E06, E0X, E12, E16, E18, E22, E26, E32, E38, E3X, E42, E46, E4X, E52, E54, E56, E5X, E62, E72, E76, E7X, E82, E84, E86, E88, E8X, E92, E98, E9X, EX2, EX6, EXX, EE2, EEX, ...

An even nontotient may be one more than a prime number, but never one less, since all numbers below a prime number are, by definition, coprime to it. To put it algebraically, for p prime: φ(p) = p − 1. Also, a pronic number n(n − 1) is certainly not a nontotient if n is prime since φ(p2) = p(p − 1).

If a natural number n is a totient, it can be shown that n×2k is a totient for all natural number k.

There are infinitely many even nontotient numbers: indeed, there are infinitely many distinct primes p such that all numbers of the form 2ap are nontotient, and every odd number has an even multiple which is a nontotient.

Most numbers end with 2 are nontotients, except 2 itself, the first counterexample is 92 = φ(X1) = φ(182), in fact, all numbers end with 2 not of the form (p−1)pk with p prime and k ≥ 1 are nontotients, (see OEIS sequence A063668 for the counterexamples) (besides, numbers end with 2 except 2 itself cannot be totients of squarefree numbers), besides, all even nontotients < 58 except 2X end with 2.

nnumbers k such that φ(k) = nnnumbers k such that φ(k) = nnnumbers k such that φ(k) = nnnumbers k such that φ(k) = n
11, 2, 316191
23, 4, 6, 326292X1, 182,
3336393
45, 8, X, 10, 3435, 47, 63, 6X, 74, 84, 92, E0, 106, 649495, 101, 16X, 174, 202, 250,
5356595
67, 9, 12, 16, 3637, 41, 72, 82, 6667, 112, 96
7376797
813, 14, 18, 20, 26, 3859, 78, E6, 68X3, 118, 119, 128, 148, 164, 186, 1X0, 210, 236, 98129, 178, 256,
9396999
XE, 1X, 3X3E, 7X, 6X6E, 11X, 9X
E3E6E9E
1011, 19, 22, 24, 30, 36, 4055, 88, 89, 94, XX, E8, 100, 110, 120, 130, 156, 70X9, 103, 124, 144, 196, 206, X0EE, 10E, 127, 133, 169, 173, 184, 188, 1EX, 218, 21X, 252, 266, 270, 290, 316, 326,
114171X1
124272X2
134373X3
1415, 28, 2X, 34, 40, 50, 4445, 8X, 7475, 97, 12X, 134, 172, 1E0, X4
154575X5
1617, 23, 32, 46, 4669, 116, 76X6X7, 192,
174777X7
1821, 29, 38, 42, 56, 4873, 98, 126, 78E9, 138, 1E6, X8193, 194, 1X8, 228, 244, 280, 2X0, 340, 366,
194979X9
1X1E, 3X, 4X4E, 9X, 7XXXXE, 19X,
1E4E7EXE
202E, 33, 39, 44, 48, 5X, 60, 66, 70, 76, 5051, 65, 79, 83, X2, X4, 10X, 136, 146, 8081, 9E, 109, 142, 143, 154, 168, 17X, 198, 1E4, 200, 216, 220, 240, 260, 286, 2E0, E0115, 149, 153, 1X4, 22X, 296, 2X6,
215181E1
225282E2
235383E3
2425, 4X, 5471, X8, E4, 114, 122, 140, 150, 180, 8485, X5, 14X, 18X, E4E5, 1XX,
255585E5
2627, 52, 5657, E2, 8687, 152, E6E7, 1E2,
275787E7
2843, 54, 58, 68, 80, 86, X0, 5888113, 158, 226, E8159, 1E8, 2E6,
295989E9
2X5X5E, EX, 8X8E, 15X, EX
2E5E8EEE
3031, 49, 53, 62, 64, 90, 96, X6, 6061, 77, 7E, 93, 99, E3, 102, 104, 108, 132, 13X, 160, 166, 170, 176, 190, 1X6, 9091, E1, 123, 139, 162, 1X2, 230, 246, 276, 100135, 163, 1X9, 1E9, 204, 208, 214, 223, 264, 26X, 278, 300, 306, 310, 320, 330, 360, 390, 396, 3E6, 446,
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