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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
OFFSET
1,2
LINKS
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
KEYWORD
nonn
AUTHOR
Labos Elemer, Oct 18 2001
STATUS
approved