cp's OEIS Frontend

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

%I A065146 #22 Jun 18 2018 03:37:40
%S A065146 1,248,264,418,477,1485,3080,3135,3596,3828,5396,10098,12648,20026,
%T A065146 21318,22152,23374,24882,35074,35343,39105,41656,44660,49938,55154,
%U A065146 56536,61344,71145,74613,86304,87087,104931,118296,124605,129504,130356,147560,150195
%N A065146 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and sigma(n) are all integers.
%H A065146 Harry J. Smith and Donovan Johnson, <a href="/A065146/b065146.txt">Table of n, a(n) for n = 1..5000</a> (first 500 terms from Harry J. Smith)
%F A065146 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().
%e A065146 n = 5396, phi(n) = 2520, sigma(n) = 10080, arithmetic mean = 6300, geometric mean = 5040, harmonic mean = 4032; 67 cases < 10^6.
%t A065146 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}]
%o A065146 (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
%Y A065146 Cf. A000010, A000203, A062354, A011257.
%K A065146 nonn
%O A065146 1,2
%A A065146 _Labos Elemer_, Oct 18 2001