

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

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. 5357 (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

Cf. A000010, A003401, A003586, A058383.
Sequence in context: A102306 A066962 A067020 * A180645 A196091 A222945
Adjacent sequences: A051910 A051911 A051912 * A051914 A051915 A051916


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



