OFFSET
1,1
COMMENTS
Also numbers n such that phi(n) = 2^a*3^b with a, b > 0.
Also numbers n such that a regular n-gon can be constructed using conics but not with merely a compass and straightedge.
"Constructed using conics" means that one can draw any conic, once its focus, its vertex and a point on its directrix are constructed. Points at intersections are thereby constructed. (Videla's definition is slightly more complicated, but equivalent.) One can use parabolas to take cube roots; hyperbolas yield trisected angles. - Don Reble, Apr 23 2007
REFERENCES
George E. Martin, Geometric Constructions, Springer, 1997, p. 140.
LINKS
T. D. Noe, Table of n, a(n) for n=1..1000
C. R. Videla, On points constructible from conics, Mathematical Intelligencer, 19, No. 2, pp. 53-57 (1997).
FORMULA
Numbers n of the form 2^a*3^b*p*q*r*..., where p, q, r, ... are distinct primes of the form 2^x*3^y + 1 (i.e., belong to A005109) and phi(n) is not a power of 2 [Videla]. - Robert G. Wilson v, Apr 05 2005
EXAMPLE
Phi(999) = Phi(3*3*3*37) = 648 = 8*81.
MATHEMATICA
lf[x_] := Length[FactorInteger[x]] eu[x_] := EulerPhi[x] Do[s=lf[eu[n]]; If[Equal[s, 2]&&Equal[Mod[eu[n], 6], 0], Print[n]], {n, 1, 1000}] (* Labos Elemer, Dec 28 2001 *)
f[n_] := Block[{m = n}, If[m > 0, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; m == 1]; fQ[n_] := Block[{pff = Select[ FactorInteger[n], #[[1]] > 3 &]}, pf = Flatten[{2, Table[ #[[1]], {1}] & /@ pff}]; pfe = Union[ Flatten[{1, Table[ #[[2]], {1}] & /@ pff}]]; If[ Union[f /@ (pf - 1)] == {True} && pfe == {1} && !IntegerQ[ Log[2, EulerPhi[ n]]], True, False]]; Select[ Range[184], fQ[ # ] &] (* Robert G. Wilson v, Apr 05 2005 *)
PROG
(Magma) [n: n in [1..200] | EulerPhi(n)/EulerPhi(EulerPhi(n)) eq 3]; // Vincenzo Librandi, Apr 17 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
J. H. Conway and Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Dec 17 1999
EXTENSIONS
Additional comments from Labos Elemer, Dec 28 2001
Additional comments from Benoit Cloitre, Jan 26 2002
Edited by N. J. A. Sloane, Apr 21 2007
Entries checked by Don Reble, Apr 23 2007
STATUS
approved