A051913 Numbers k such that phi(k)/phi(phi(k)) = 3.

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

Offset: 1

Author

J. H. Conway and Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Dec 17 1999

Keywords

• nonn
• easy

Comments

Also numbers k such that phi(k) = 2^a*3^b with a, b > 0.

Also numbers k such that a regular k-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

Examples

			Phi(999) = Phi(3*3*3*37) = 648 = 8*81.
		

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).

Crossrefs

Cf. A000010, A003401, A003586, A058383.

Programs

Magma

[n: n in [1..200] | EulerPhi(n)/EulerPhi(EulerPhi(n)) eq 3]; // Vincenzo Librandi, Apr 17 2015

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 *)
Select[Range[200],EulerPhi[#]/EulerPhi[EulerPhi[#]]==3&] (* Harvey P. Dale, Jul 11 2025 *)

Python

from itertools import count, islice
from sympy import primefactors, totient
def A051913_gen(): # generator of terms
    yield from filter(lambda n: primefactors(totient(n)) == [2,3], count(1))
A051913_list = list(islice(A051913_gen(),30)) # Chai Wah Wu, Apr 02 2025

Formula

Numbers k 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(k) is not a power of 2 [Videla]. - Robert G. Wilson v, Apr 05 2005

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