A291959 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and psi(n) are all integers, where phi(n) is the Euler totient function (A000010) and psi(n) is the Dedekind psi function (A001615).

1, 150, 300, 418, 450, 525, 600, 750, 836, 900, 1200, 1350, 1500, 1575, 1672, 1800, 2250, 2400, 2625, 2700, 3000, 3135, 3344, 3600, 3675, 3750, 4050, 4500, 4598, 4725, 4800, 5400, 6000, 6688, 6750, 7200, 7500, 7875, 7942, 8100, 9000, 9196, 9405, 9600, 10800

Offset

1,2

Comments

The number of terms below 10^3, 10^4, ... are 10, 44, 147, 397, ...

Links

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

Example

phi(150)=40, psi(150)=360, their arithmetic mean = 200, geometric mean = 120, harmonic mean = 72 are all integers.

Mathematica

dedekindPsi[n_] := If[n < 1, 0, n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]];
aQ[n_] := Module[{s = dedekindPsi[n], p = EulerPhi[n]}, IntegerQ[(s + p)/2] && IntegerQ[Sqrt[s*p]] && IntegerQ[2 s*p/(s + p)]]; Select[Range[10^5], aQ]

Crossrefs

Cf. A000010, A001615, A065146.

Sequence in context: A292705 A335145 A063829 * A210643 A211550 A212465

Adjacent sequences: A291956 A291957 A291958 * A291960 A291961 A291962

Keyword

nonn

Author

Amiram Eldar, Sep 06 2017

Status

approved