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

%I A051913 #42 Jul 12 2025 16:11:49
%S A051913 7,9,13,14,18,19,21,26,27,28,35,36,37,38,39,42,45,52,54,56,57,63,65,
%T A051913 70,72,73,74,76,78,81,84,90,91,95,97,104,105,108,109,111,112,114,117,
%U A051913 119,126,130,133,135,140,144,146,148,152,153,156,162,163,168,171,180,182
%N A051913 Numbers k such that phi(k)/phi(phi(k)) = 3.
%C A051913 Also numbers k such that phi(k) = 2^a*3^b with a, b > 0.
%C A051913 Also numbers k such that a regular k-gon can be constructed using conics but not with merely a compass and straightedge.
%C A051913 "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
%D A051913 George E. Martin, Geometric Constructions, Springer, 1997, p. 140.
%H A051913 T. D. Noe, <a href="/A051913/b051913.txt">Table of n, a(n) for n=1..1000</a>
%H A051913 C. R. Videla, <a href="http://dx.doi.org/10.1007/BF03024434">On points constructible from conics</a>, Mathematical Intelligencer, 19, No. 2, pp. 53-57 (1997).
%F A051913 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
%e A051913 Phi(999) = Phi(3*3*3*37) = 648 = 8*81.
%t A051913 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 *)
%t A051913 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 *)
%t A051913 Select[Range[200],EulerPhi[#]/EulerPhi[EulerPhi[#]]==3&] (* _Harvey P. Dale_, Jul 11 2025 *)
%o A051913 (Magma) [n: n in [1..200] | EulerPhi(n)/EulerPhi(EulerPhi(n)) eq 3]; // _Vincenzo Librandi_, Apr 17 2015
%o A051913 (Python)
%o A051913 from itertools import count, islice
%o A051913 from sympy import primefactors, totient
%o A051913 def A051913_gen(): # generator of terms
%o A051913     yield from filter(lambda n: primefactors(totient(n)) == [2,3], count(1))
%o A051913 A051913_list = list(islice(A051913_gen(),30)) # _Chai Wah Wu_, Apr 02 2025
%Y A051913 Cf. A000010, A003401, A003586, A058383.
%K A051913 nonn,easy
%O A051913 1,1
%A A051913 _J. H. Conway_ and Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Dec 17 1999
%E A051913 Additional comments from _Labos Elemer_, Dec 28 2001
%E A051913 Additional comments from _Benoit Cloitre_, Jan 26 2002
%E A051913 Edited by _N. J. A. Sloane_, Apr 21 2007
%E A051913 Entries checked by _Don Reble_, Apr 23 2007