A065146 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and sigma(n) are all integers.

1, 248, 264, 418, 477, 1485, 3080, 3135, 3596, 3828, 5396, 10098, 12648, 20026, 21318, 22152, 23374, 24882, 35074, 35343, 39105, 41656, 44660, 49938, 55154, 56536, 61344, 71145, 74613, 86304, 87087, 104931, 118296, 124605, 129504, 130356, 147560, 150195

Offset: 1

Author

Labos Elemer, Oct 18 2001

Keywords

• nonn

Examples

			n = 5396, phi(n) = 2520, sigma(n) = 10080, arithmetic mean = 6300, geometric mean = 5040, harmonic mean = 4032; 67 cases < 10^6.
		

Links

• Harry J. Smith and Donovan Johnson, Table of n, a(n) for n = 1..5000 (first 500 terms from Harry J. Smith)

Crossrefs

Cf. A000010, A000203, A062354, A011257.

Programs

Mathematica

Do[s = EulerPhi[n]*DivisorSigma[1, n]; z = (EulerPhi[n]+DivisorSigma[1, n])/2; u = h[n]; If[IntegerQ[Sqrt[s]]&&IntegerQ[z]&&IntegerQ[u], Print[n]], {n, 1, 1000000}]

PARI

{ n=0; for (m=1, 10^9, e=eulerphi(m); s=sigma(m); if (!issquare(e*s), next); h=(2*e*s)/(e + s); if (frac(h) != 0, next); if (frac((e + s)/2) != 0, next); write("b065146.txt", n++, " ", m); if (n==500, return) ) } \\ Harry J. Smith, Oct 12 2009

Formula

a = (phi(n)+sigma(n))/2, g = sqrt(phi(n)*sigma(n)), h = (2*phi(n)*sigma(n))/(phi(n)+sigma(n)) = g^2/a are all integers; phi() = A000010(), sigma() = A000203().