Nontotient


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 = 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
Least k such that the totient of k is n are
Greatest k such that the totient of k is n are
Number of ks such that φ = n are
According to Carmichael's conjecture there are no 1's in this sequence.
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 − 1. Also, a pronic number n is certainly not a nontotient if n is prime since φ = p.
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.
nnumbers k such that φ = nnnumbers k such that φ = nnnumbers k such that φ = nnnumbers k such that φ = n
11, 23773109
23, 4, 63874110121, 242
33975111
45, 8, 10, 124041, 55, 75, 82, 88, 100, 110, 132, 15076112113, 145, 226, 232, 290, 348
54177113
67, 9, 14, 184243, 49, 86, 987879, 158114
74379115
815, 16, 20, 24, 304469, 92, 13880123, 164, 165, 176, 200, 220, 246, 264, 300, 330116177, 236, 354
94581117
1011, 224647, 948283, 166118
114783119
1213, 21, 26, 28, 36, 424865, 104, 105, 112, 130, 140, 144, 156, 168, 180, 21084129, 147, 172, 196, 258, 294120143, 155, 175, 183, 225, 231, 244, 248, 286, 308, 310, 350, 366, 372, 396, 450, 462
134985121
145086122
155187123
1617, 32, 34, 40, 48, 605253, 1068889, 115, 178, 184, 230, 276124
175389125
1819, 27, 38, 545481, 16290126127, 254
195591127
2025, 33, 44, 50, 665687, 116, 17492141, 188, 282128255, 256, 272, 320, 340, 384, 408, 480, 510
215793129
2223, 465859, 11894130131, 262
235995131
2435, 39, 45, 52, 56, 70, 72, 78, 84, 906061, 77, 93, 99, 122, 124, 154, 186, 1989697, 119, 153, 194, 195, 208, 224, 238, 260, 280, 288, 306, 312, 336, 360, 390, 420132161, 201, 207, 268, 322, 402, 414
256197133
266298134
276399135
2829, 586485, 128, 136, 160, 170, 192, 204, 240100101, 125, 202, 250136137, 274
2965101137
3031, 626667, 134102103, 206138139, 278
3167103139
3251, 64, 68, 80, 96, 102, 12068104159, 212, 318140213, 284, 426
3369105141
347071, 142106107, 214142
3571107143
3637, 57, 63, 74, 76, 108, 114, 1267273, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270108109, 133, 171, 189, 218, 266, 324, 342, 378144185, 219, 273, 285, 292, 296, 304, 315, 364, 370, 380, 432, 438, 444, 456, 468, 504, 540, 546, 570, 630