A002180 - OEIS

%I M0500 N0180 #42 May 08 2018 15:11:53

%S 1,2,3,4,5,6,8,9,10,11,12,14,15,16,18,20,21,22,23,24,26,27,28,29,30,

%T 32,33,35,36,39,40,41,42,44,46,48,50,51,52,53,54,55,56,58,60,63,64,65,

%U 66,68,69,70,72,74,75,78,80,81,82,83,84,86,88,89,90,92

%N Values taken by the half-totient function phi(m)/2.

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

%D J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%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="/A002180/b002180.txt">Table of n, a(n) for n = 2..10000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H R. D. Carmichael, <a href="/A002180/a002180.pdf">A table of the values of m corresponding to given values of phi(m)</a>, Amer. J. Math., 30 (1908),394-400. [Annotated scanned copy]

%H K. W. Wegner, <a href="/A002180/a002180_1.pdf">Values of phi(x) = n for n from 2 through 1978</a>, mimeographed manuscript, no date [Annotated scanned copy]

%F a(n) = A002202(n)/2 for n > 1.

%p with(numtheory); t1 := [seq(nops(invphi(n)), n=1..300)]; t2 := []: for n from 2 to 300 do if t1[n] <> 0 then t2 := [op(t2), n/2]; fi; od: t2;

%t phiQ[m_] := Select[Range[m+1, 2 m*Product[(1-1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; Select[Range[2, 200], phiQ]/2 (* _Jean-François Alcover_, Jun 13 2012, after Maxim Rytin *)

%o (Haskell)

%o a002180 = flip div 2 . a002202  -- _Reinhard Zumkeller_, Nov 22 2015

%o (PARI) list(lim)=my(v=List()); for(n=1,lim, if(istotient(2*n), listput(v,n))); Vec(v) \\ _Charles R Greathouse IV_, Feb 08 2017

%Y Cf. A002202, A079695 (complementary sequence).

%K nonn,nice,easy

%O 2,2

%A _N. J. A. Sloane_

%E More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001