

A005277


Nontotients: even n such that phi(m) = n has no solution.
(Formerly M4927)


87



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

1,1


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 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


REFERENCES

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).


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
Lambert A'Campo, Every 7Dimensional Abelian Variety over the padic Numbers has a Reducible Ladic Galois Representation, arXiv:2006.06737 [math.NT], 2020.
Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, arXiv:1801.04542 [hepth], 2018.
K. Ford, S. Konyagin, and C. Pomerance, Residue classes free of values of Euler's function (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


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


EXAMPLE

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


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 *)


PROG

(Haskell)
a005277 n = a005277_list !! (n1)
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


CROSSREFS

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


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Jud McCranie, Oct 13 2000


STATUS

approved



