In number theory, a nontotient is a positive integer n which is not a totient number: it is not in the image 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 this sequence:

14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, ...

The least value of k such that the totient of k is n are (0 if no such k exists) are this sequence: 1, 3, 0, 5, 0, 7, 0, 15, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 35, 0, 0, 0, 29, 0, 31, 0, 51, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 69, 0, 47, 0, 65, 0, 0, 0, 53, 0, 81, 0, 87, 0, 59, 0, 61, 0, 0, 0, 85, 0, 67, 0, 0, 0, 71, 0, 73, ...

The greatest value of k such that the totient of k is n are (0 if no such k exists) are this sequence: 2, 6, 0, 12, 0, 18, 0, 30, 0, 22, 0, 42, 0, 0, 0, 60, 0, 54, 0, 66, 0, 46, 0, 90, 0, 0, 0, 58, 0, 62, 0, 120, 0, 0, 0, 126, 0, 0, 0, 150, 0, 98, 0, 138, 0, 94, 0, 210, 0, 0, 0, 106, 0, 162, 0, 174, 0, 118, 0, 198, 0, 0, 0, 240, 0, 134, 0, 0, 0, 142, 0, 270, ...

The number of ks such that φ(k) = n are (start with n = 0) are this sequence: 0, 2, 3, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, 0, 0, 0, 6, 0, 4, 0, 5, 0, 2, 0, 10, 0, 0, 0, 2, 0, 2, 0, 7, 0, 0, 0, 8, 0, 0, 0, 9, 0, 4, 0, 3, 0, 2, 0, 11, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 9, 0, 0, 0, 8, 0, 2, 0, 0, 0, 2, 0, 17, ... Carmichael's conjecture is that there are no 1s 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) = 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, n · 2k is a totient for all natural numbers k.

There are infinitely many even nontotient numbers: indeed, there are infinitely many distinct primes p (such as 78557 and 271129, see Sierpinski number) 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(k) = nnnumbers k such that φ(k) = nnnumbers k such that φ(k) = nnnumbers k such that φ(k) = n11, 2377310923, 4, 63874110121, 2423397511145, 8, 10, 124041, 55, 75, 82, 88, 100, 110, 132, 15076112113, 145, 226, 232, 290, 3485417711367, 9, 14, 184243, 49, 86, 987879, 15811474379115815, 16, 20, 24, 304469, 92, 13880123, 164, 165, 176, 200, 220, 246, 264, 300, 330116177, 236, 354945811171011, 224647, 948283, 1661181147831191213, 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, 4621349851211450861221551871231617, 32, 34, 40, 48, 605253, 1068889, 115, 178, 184, 230, 2761241753891251819, 27, 38, 545481, 16290126127, 2541955911272025, 33, 44, 50, 665687, 116, 17492141, 188, 282128255, 256, 272, 320, 340, 384, 408, 480, 5102157931292223, 465859, 11894130131, 2622359951312435, 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, 4142561971332662981342763991352829, 586485, 128, 136, 160, 170, 192, 204, 240100101, 125, 202, 250136137, 27429651011373031, 626667, 134102103, 206138139, 27831671031393251, 64, 68, 80, 96, 102, 12068104159, 212, 318140213, 284, 4263369105141347071, 142106107, 21414235711071433637, 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

References

L. Havelock, A Few Observations on Totient and Cototient Valence from PlanetMath