A083546 The geometric mean of the Euler totient function of 2 consecutive integers {k, k+1} when it is an integer.

1, 2, 8, 12, 48, 48, 60, 80, 96, 128, 144, 180, 280, 240, 240, 288, 288, 288, 336, 288, 384, 360, 480, 480, 640, 720, 672, 600, 576, 720, 720, 720, 672, 864, 960, 864, 960, 1080, 1008, 1408, 1296, 960, 1008, 1320, 1260, 1056, 1440, 1200, 1728, 1440, 1296

Offset: 1

Author

Labos Elemer, May 21 2003

Keywords

• nonn

Examples

			12 is a term since sqrt(phi(19) * phi(20)) = sqrt(18 * 8) = sqrt(144) = 12.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A083542, A083542, A066813, A058515, A083542, A083545.

Programs

Mathematica

f[x_] := EulerPhi[x]; Do[s=Sqrt[f[n+1]*f[n]]; If[IntegerQ[s], Print[s]], {n, 1, 5000}]
Select[Sqrt[#]&/@(Times@@@Partition[EulerPhi[Range[3000]],2,1]),IntegerQ] (* Harvey P. Dale, Nov 04 2020 *)

Formula

a(n) = sqrt(A000010(x) * A000010(x+1)) = sqrt(phi(x) * phi(x+1)) = sqrt(A083542(x)) where x = A083545(n).