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A348868
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Numbers whose numerator and denominator of the harmonic mean of their divisors are both 5-smooth numbers.
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2
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1, 2, 3, 5, 6, 8, 10, 15, 24, 27, 28, 30, 40, 54, 84, 120, 135, 140, 216, 224, 270, 420, 496, 672, 756, 775, 819, 1080, 1120, 1488, 1550, 1638, 2176, 2325, 2480, 3360, 3780, 4095, 4650, 6048, 6200, 6528, 6552, 7440, 8190, 10880, 11375, 13392, 18600, 20925, 21700
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OFFSET
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1,2
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COMMENTS
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The terms that are also harmonic numbers (A001599) are those whose harmonic mean of divisors (A001600) is a 5-smooth number. Of the 937 harmonic numbers below 10^14, 83 are terms in this sequence.
If k1 and k2 are coprime terms, then k1*k2 is also a term. In particular, if k is an odd term, then 2*k is also a term.
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LINKS
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EXAMPLE
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8 is a term since the harmonic mean of its divisors is 32/15 and both 32 = 2^5 and 15 = 3*5 are 5-smooth numbers.
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MATHEMATICA
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smQ[n_] := n == 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3] * 5^IntegerExponent[n, 5]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := smQ[Numerator[(hn = h[n])]] && smQ[Denominator[hn]]; Select[Range[22000], q]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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