

A051913


Numbers n such that phi(n)/phi(phi(n)) = 3.


4



7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 35, 36, 37, 38, 39, 42, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97, 104, 105, 108, 109, 111, 112, 114, 117, 119, 126, 130, 133, 135, 140, 144, 146, 148, 152, 153, 156, 162, 163, 168, 171, 180, 182
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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 ngon 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



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



STATUS

approved



