A083545 Numbers k such that the geometric mean of the Euler totient function of k and k+1 is an integer.

1, 3, 15, 19, 95, 104, 125, 164, 194, 255, 259, 341, 491, 495, 504, 512, 513, 584, 591, 629, 679, 755, 775, 975, 1024, 1147, 1247, 1254, 1260, 1313, 1358, 1463, 1469, 1538, 1615, 1728, 1919, 1962, 1970, 2047, 2071, 2090, 2204, 2299, 2321, 2345, 2404, 2625

Offset: 1

Author

Labos Elemer, May 21 2003

Keywords

• nonn

Examples

			19 is a term since phi(19) = 18, phi(20) = 8, 8*18 = 144 = 12^2.
		

Links

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

Crossrefs

Cf. A000010, A083542, A066813, A058515, A083538, A083542, A083546.

Programs

Mathematica

f[x_] := EulerPhi[x]; Do[s=Sqrt[f[n+1]*f[n]]; If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Position[Partition[EulerPhi[Range[2700]],2,1],?(IntegerQ[GeometricMean[ #]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 13 2020 *)

Formula

a(n) = x is such that sqrt(A000010(x)*A000010(x+1)) is an integer. Values of solutions x to phi(x) * phi(x+1) = A083542(x) = y^2.