A002180 Values taken by the half-totient function phi(m)/2. (Formerly M0500 N0180)

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, 32, 33, 35, 36, 39, 40, 41, 42, 44, 46, 48, 50, 51, 52, 53, 54, 55, 56, 58, 60, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 78, 80, 81, 82, 83, 84, 86, 88, 89, 90, 92

Offset

2,2

References

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.

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

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

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 = 2..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

R. D. Carmichael, A table of the values of m corresponding to given values of phi(m), Amer. J. Math., 30 (1908),394-400. [Annotated scanned copy]

K. W. Wegner, Values of phi(x) = n for n from 2 through 1978, mimeographed manuscript, no date [Annotated scanned copy]

Formula

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

Maple

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;

Mathematica

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 *)

Prog

(Haskell)
a002180 = flip div 2 . a002202  -- Reinhard Zumkeller, Nov 22 2015
(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

Crossrefs

Cf. A002202, A079695 (complementary sequence).

Sequence in context: A030173 A048265 A288712 * A207333 A179182 A298303

Adjacent sequences: A002177 A002178 A002179 * A002181 A002182 A002183

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved