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%I M4927 #159 Mar 22 2023 08:08:11
%S 14,26,34,38,50,62,68,74,76,86,90,94,98,114,118,122,124,134,142,146,
%T 152,154,158,170,174,182,186,188,194,202,206,214,218,230,234,236,242,
%U 244,246,248,254,258,266,274,278,284,286,290,298,302,304,308,314,318
%N Nontotients: even numbers k such that phi(m) = k has no solution.
%C 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
%C 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
%C 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
%D R. K. Guy, Unsolved Problems in Number Theory, B36.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert G. Wilson v, <a href="/A005277/b005277.txt">Table of n, a(n) for n = 1..29750</a> (terms 1..10000 from T. D. Noe).
%H Lambert A'Campo, <a href="https://arxiv.org/abs/2006.06737">Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible L-adic Galois Representation</a>, arXiv:2006.06737 [math.NT], 2020.
%H Matteo Caorsi and Sergio Cecotti, <a href="https://arxiv.org/abs/1801.04542">Geometric classification of 4d N=2 SCFTs</a>, arXiv:1801.04542 [hep-th], 2018.
%H K. Ford, S. Konyagin, and C. Pomerance, <a href="https://arxiv.org/abs/2005.01078">Residue classes free of values of Euler's function</a>, arXiv:2005.01078 [math.NT] (1999).
%H L. Havelock, <a href="https://web.archive.org/web/20080703175341/http://aux.planetmath.org/files/papers/335/C:TempObsTotientCototientValence.pdf">A Few Observations on Totient and Cototient Valence</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Nontotient.html">Nontotient.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Nontotient">Nontotient</a>
%H Robert G. Wilson v, <a href="/A007015/a007015.pdf">Letter to N. J. A. Sloane, Jul. 1992</a>
%F a(n) = 2*A079695(n). - _R. J. Mathar_, Sep 29 2021
%F {k: k even and A014197(k) = 0}. - _R. J. Mathar_, Sep 29 2021
%e There are no values of m such that phi(m)=14, so 14 is a term of the sequence.
%p A005277 := n -> if type(n,even) and invphi(n)=[] then n fi: seq(A005277(i),i=1..318); # _Peter Luschny_, Jun 26 2011
%t 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 *)
%t 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 *)
%o (Haskell)
%o a005277 n = a005277_list !! (n-1)
%o a005277_list = filter even a007617_list
%o -- _Reinhard Zumkeller_, Nov 22 2015
%o (PARI) is(n)=n%2==0 && !istotient(n) \\ _Charles R Greathouse IV_, Mar 04 2017
%o (Magma) [n: n in [2..400 by 2] | #EulerPhiInverse(n) eq 0]; // _Marius A. Burtea_, Sep 08 2019
%Y See A007617 for all numbers k (odd or even) such that phi(m) = k has no solution.
%Y All even numbers not in A002202. Cf. A000010.
%Y Cf. A005384, A006093.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Jud McCranie_, Oct 13 2000
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