

A005277


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


92



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 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


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



FORMULA



EXAMPLE

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


MAPLE



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 JeanFrançois Alcover, May 23 2011 in A002202 *) Select[2 Range@160, ! totientQ@# &] (* Robert G. Wilson v, Mar 20 2023 *)


PROG

(Haskell)
a005277 n = a005277_list !! (n1)
a005277_list = filter even a007617_list
(Magma) [n: n in [2..400 by 2]  #EulerPhiInverse(n) eq 0]; // Marius A. Burtea, Sep 08 2019


CROSSREFS

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


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



