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 A005277 Nontotients: even n such that phi(m) = n has no solution. (Formerly M4927) 55

%I M4927

%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 n such that phi(m) = n 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 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

%C 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 Sierpinski numbers. This shows that 271129*2^k is a nontotient for all k>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 T. D. Noe, <a href="/A005277/b005277.txt">Table of n, a(n) for n=1..10000</a>

%H K. Ford, S. Konyagin, C. Pomerance, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.123.4370">Residue classes free of values of Euler's function</a> (1999). [From _R. J. Mathar_, Mar 15 2010]

%H L. Havelock, <a href="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>

%e There are no values of m such that phi(m)=14, so 14 is a member 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 Complement[2*Range[159], Flatten[{Table[(Prime[i] - 1)*(Prime[j] - 1), {i, 70}, {j, 70}], Table[(Prime[i] - 1)*(Prime[j]^i), {i, 70}, {j, 8}]}]] (* _Alonso del Arte_, Jun 10 2006 *)

%Y See A007617 for all values. Cf. A000010.

%Y Cf. A005384.

%Y Cf. A006093.

%K nonn

%O 1,1

%A _N. J. A. Sloane_.

%E More terms from _Jud McCranie_, Oct 13 2000

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