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

1,2

Links

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

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().

Example

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

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}]

Prog

(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

Crossrefs

Cf. A000010, A000203, A062354, A011257.

Sequence in context: A048901 A223546 A187398 * A064977 A068400 A240967

Adjacent sequences: A065143 A065144 A065145 * A065147 A065148 A065149

Keyword

nonn

Author

Labos Elemer, Oct 18 2001

Status

approved