A065150 Numbers k such that the harmonic mean of phi(k) and sigma(k) is an integer.

1, 12, 15, 35, 56, 78, 95, 140, 143, 172, 190, 248, 264, 287, 315, 319, 323, 357, 418, 477, 588, 594, 675, 812, 814, 840, 899, 910, 1045, 1107, 1118, 1131, 1199, 1208, 1254, 1349, 1420, 1425, 1485, 1495, 1558, 1608, 1672, 1763, 2214, 2261, 2318, 2337

Offset: 1

Author

Labos Elemer, Oct 18 2001

Keywords

• nonn

Examples

			m = 319, phi(319) = 280, sigma(319) = 360; phi(319)*sigma(319) = 100800, phi(319) + sigma(319) = 640; 1/(harmonic mean) = (640/100800)/2, harmonic mean = 315, arithmetic mean = 320, geometric mean is not an integer.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Crossrefs

Cf. A000010, A000203, A065146, A011257.

Programs

Mathematica

Select[Range[2400], IntegerQ[HarmonicMean @ {EulerPhi[#], DivisorSigma[1, #]}] &] (* Amiram Eldar, Mar 20 2025 *)

PARI

{ n=0; for (m=1, 10^9, e=eulerphi(m); s=sigma(m); h=(2*e*s)/(e + s); if (frac(h) == 0, write("b065150.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 13 2009

Formula

G^2 mod A = 0, where G^2 = A000010(m)*A000203(m), A = (A000010(m) + A000203(m))/2; harmonic mean = (G^2)/A is an integer.