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A051913
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Numbers n such that phi(n)/phi(phi(n))=3.
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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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
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REFERENCES
| George E. Martin, Geometric Constructions, Springer, 1997, p. 140.
C. R. Videla, On points constructible from conics, Mathematical Intelligencer, 19, No. 2, pp. 53-57 (1997).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
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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 (rgwv(AT)rgwv.com), Apr 05 2005
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EXAMPLE
| Phi[999]=Phi[3*3*3*37]=648=8*81.
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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 E. (labos(AT)ana.sote.hu), 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[ # ] &] (from Robert G. Wilson v Apr 05 2005)
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CROSSREFS
| Cf. A000010, A003401, A003586, A058383.
Sequence in context: A102306 A066962 A067020 * A180645 A196091 A129069
Adjacent sequences: A051910 A051911 A051912 * A051914 A051915 A051916
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KEYWORD
| nonn,easy
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AUTHOR
| John Conway (conway(AT)math.Princeton.EDU) and Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Dec 17 1999
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EXTENSIONS
| Additional comments from Labos E. (labos(AT)ana.sote.hu), Dec 28 2001 and Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 26 2002
Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 21 2007
Entries checked by Don Reble (djr(AT)nk.ca), Apr 23 2007
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