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A140798
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Harmonic numbers that are not multiply-perfect.
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3
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140, 270, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000, 2845800, 4358600
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OFFSET
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1,1
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COMMENTS
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Multiply-perfect numbers m (with sigma(m)/m an integer) are necessarily harmonic numbers (with tau(m)/{sigma(m)/m } an integer), but the converse is not true : If m divides sigma(m), then quotient sigma(m)/m divides tau(m) [m=A007691]; However, quotient tau(n)/{sigma(n)/n} being an integer does not imply quotient sigma(n)/n is necessarily an integer [n=A001599].
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REFERENCES
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J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 140, pp 48, Ellipses, Paris 2008.
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LINKS
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MATHEMATICA
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Select[Range[10^6], (d = Divisors[#]; IntegerQ[HarmonicMean[d]] && Mod[Total[d], #] != 0)&] (* Jean-François Alcover, Nov 13 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Initial term 1 removed and offset corrected by Donovan Johnson, Jan 25 2012
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STATUS
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approved
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