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A065146 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and sigma(n) are all integers. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 17 03:44 EDT 2021. Contains 343059 sequences. (Running on oeis4.)