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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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]

Crossrefs

Cf. A002202, A079695 (complementary sequence).

Programs

Haskell

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

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 *)

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

Formula

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

Extensions

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