

A065150


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


1



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
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OFFSET

1,2


LINKS

Harry J. Smith, Table of n, a(n) for n = 1..1000


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 integer.


EXAMPLE

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.


PROG

(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


CROSSREFS

Cf. A000010, A000203, A065146, A011257.
Sequence in context: A194234 A296796 A161917 * A277082 A087098 A109315
Adjacent sequences: A065147 A065148 A065149 * A065151 A065152 A065153


KEYWORD

nonn


AUTHOR

Labos Elemer, Oct 18 2001


STATUS

approved



