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Innumber theory, anontotient is a positive integern which is not atotient number: it is not in theimage ofEuler's totient function φ, that is, the equation φ(x) =n has no solutionx. In other words,n is a nontotient if there is no integerx that has exactlyncoprimes below it. All odd numbers are nontotients, except1, since it has the solutionsx = 1 andx = 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, ... (sequenceA005277 in theOEIS)

The least value ofk such that the totient ofk isn are (0 if no suchk 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, ... (sequenceA049283 in theOEIS)

The greatest value ofk such that the totient ofk isn are (0 if no suchk 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, ... (sequenceA057635 in theOEIS)

The number ofks such that φ(k) =n are (start withn = 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, ... (sequenceA014197 in theOEIS)

Carmichael's conjecture is that there are no1s in this sequence.

An even nontotient may be one more than aprime 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, apronic numbern(n − 1) is certainly not a nontotient ifn is prime since φ(p²) =p(p − 1).

If a natural numbern is a totient,n · 2^k is a totient for all natural numbersk.

There are infinitely many even nontotient numbers: indeed, there are infinitely many distinct primesp (such as 78557 and 271129, seeSierpinski number) such that all numbers of the form 2^ap are nontotient, and every odd number has an even multiple which is a nontotient.

• Guy, Richard K. (2004).Unsolved Problems in Number Theory. Problem Books in Mathematics. New York, NY:Springer-Verlag. p. 139.ISBN 0-387-20860-7.Zbl 1058.11001.
• L. Havelock,A Few Observations on Totient and Cototient Valence fromPlanetMath
• Sándor, Jozsef; Crstici, Borislav (2004).Handbook of number theory II. Dordrecht: Kluwer Academic. p. 230.ISBN 1-4020-2546-7.Zbl 1079.11001.
• Zhang, Mingzhi (1993)."On nontotients".Journal of Number Theory.43 (2):168–172.doi:10.1006/jnth.1993.1014.ISSN 0022-314X.Zbl 0772.11001.

• Integer sequences

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