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A005277
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Nontotients: even n such that phi(m) = n has no solution.
(Formerly M4927)
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87
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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
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OFFSET
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1,1
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COMMENTS
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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 n^2 + 1 such that n^2 + 2 is composite. These n are given in A106571. Similarly, let b be 3 or a number such that b=1 (mod 4). For any k > 0 such that b^k + 2 is composite, b^k + 1 is a nontotient. - T. D. Noe, Sep 13 2007
The Firoozbakht comment can be generalized: Observe that if n is a nontotient and 2n+1 is composite, then 2n is also a nontotient. See A057192 and A076336 for a connection to Sierpiński numbers. This shows that 271129*2^k is a nontotient for all k > 0. - T. D. Noe, Sep 13 2007
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
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
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FORMULA
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a(n) = 2*A079695(n). - R. J. Mathar, Sep 29 2021
{k: k even and A014197(k) =0}. - R. J. Mathar, Sep 29 2021
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EXAMPLE
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There are no values of m such that phi(m)=14, so 14 is a member of the sequence.
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MAPLE
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A005277 := n -> if type(n, even) and invphi(n)=[] then n fi: seq(A005277(i), i=1..318); # Peter Luschny, Jun 26 2011
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a005277 n = a005277_list !! (n-1)
a005277_list = filter even a007617_list
-- Reinhard Zumkeller, Nov 22 2015
(PARI) is(n)=n%2==0 && !istotient(n) \\ Charles R Greathouse IV, Mar 04 2017
(Magma) [n: n in [2..400 by 2] | #EulerPhiInverse(n) eq 0]; // Marius A. Burtea, Sep 08 2019
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CROSSREFS
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See A007617 for all values. All even numbers not in A002202. Cf. A000010.
Cf. A005384, A006093.
Sequence in context: A134837 A105583 A323030 * A079702 A235688 A176274
Adjacent sequences: A005274 A005275 A005276 * A005278 A005279 A005280
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Jud McCranie, Oct 13 2000
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STATUS
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approved
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