Nontotient
Number that is not in the range of Euler's totient function
title: "Nontotient" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["integer-sequences"] description: "Number that is not in the range of Euler's totient function" topic_path: "general/integer-sequences" source: "https://en.wikipedia.org/wiki/Nontotient" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Number that is not in the range of Euler's totient function ::
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 2a**p are nontotient, and every odd number has an even multiple which is a nontotient.
::data[format=table]
| 36 | 37, 57, 63, 74, 76, 108, 114, 126 | 72 | 73, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270 | 108 | 109, 133, 171, 189, 218, 266, 324, 342, 378 | 144 | 185, 219, 273, 285, 292, 296, 304, 315, 364, 370, 380, 432, 438, 444, 456, 468, 504, 540, 546, 570, 630 |
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References
- L. Havelock, A Few Observations on Totient and Cototient Valence from PlanetMath
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