A005277 Nontotients: even numbers k such that phi(m) = k has no solution.

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, 302, 304, 308, 314, 318

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

If p is prime then the following two statements are true. I. 2p is in the sequence iff 2p+1 is composite (p is not a Sophie Germain prime). II. 4p is in the sequence iff 2p+1 and 4p+1 are composite. - Farideh Firoozbakht, Dec 30 2005

Another subset of nontotients consists of the numbers j^2 + 1 such that j^2 + 2 is composite. These numbers j are given in A106571. Similarly, let b be 3 or a number such that b == 1 (mod 4). For any j > 0 such that b^j + 2 is composite, b^j + 1 is a nontotient. - T. D. Noe, Sep 13 2007

The Firoozbakht comment can be generalized: Observe that if k is a nontotient and 2k+1 is composite, then 2k is also a nontotient. See A057192 and A076336 for a connection to Sierpiński numbers. This shows that 271129*2^j is a nontotient for all j > 0. - T. D. Noe, Sep 13 2007

Examples

			There are no values of m such that phi(m)=14, so 14 is a term of the sequence.
		

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 44 at p. 91.
• R. K. Guy, Unsolved Problems in Number Theory, B36.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 91.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..29750 (terms 1..10000 from T. D. Noe).
• Lambert A'Campo, Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible L-adic Galois Representation, arXiv:2006.06737 [math.NT], 2020.
• Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, arXiv:1801.04542 [hep-th], 2018.
• K. Ford, S. Konyagin, and C. Pomerance, Residue classes free of values of Euler's function, arXiv:2005.01078 [math.NT] (1999).
• L. Havelock, A Few Observations on Totient and Cototient Valence.
• Eric Weisstein's World of Mathematics, Nontotient.
• Wikipedia, Nontotient.
• Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Crossrefs

See A007617 for all numbers k (odd or even) such that phi(m) = k has no solution.

All even numbers not in A002202. Cf. A000010.

Cf. A005384, A006093, A014197, A057192, A076336, A079695, A106571.

Programs

Haskell

a005277 n = a005277_list !! (n-1)
a005277_list = filter even a007617_list
-- Reinhard Zumkeller, Nov 22 2015

Magma

[n: n in [2..400 by 2] | #EulerPhiInverse(n) eq 0]; // Marius A. Burtea, Sep 08 2019

Maple

A005277 := n -> if type(n,even) and invphi(n)=[] then n fi: seq(A005277(i),i=1..318); # Peter Luschny, Jun 26 2011

Mathematica

searchMax = 320; phiAnsYldList = Table[0, {searchMax}]; Do[phiAns = EulerPhi[m]; If[phiAns <= searchMax, phiAnsYldList[[phiAns]]++ ], {m, 1, searchMax^2}]; Select[Range[searchMax], EvenQ[ # ] && (phiAnsYldList[[ # ]] == 0) &] (* Alonso del Arte, Sep 07 2004 *)
totientQ[m_] := Select[ Range[m +1, 2m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; (* after Jean-François Alcover, May 23 2011 in A002202 *) Select[2 Range@160, ! totientQ@# &] (* Robert G. Wilson v, Mar 20 2023 *)

PARI

is(n)=n%2==0 && !istotient(n) \\ Charles R Greathouse IV, Mar 04 2017

Formula

a(n) = 2*A079695(n). - R. J. Mathar, Sep 29 2021

{k: k even and A014197(k) = 0}. - R. J. Mathar, Sep 29 2021

Extensions

More terms from Jud McCranie, Oct 13 2000