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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
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 06 2017
STATUS
approved