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A348867
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Numbers whose numerator and denominator of the harmonic mean of their divisors are both 3-smooth numbers.
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2
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1, 2, 3, 6, 28, 40, 84, 120, 135, 224, 270, 672, 819, 1638, 3780, 10880, 13392, 30240, 32640, 32760, 167400, 950976, 1303533, 2178540, 2607066, 3138345, 4713984, 6276690, 8910720, 14705145, 17428320, 29410290, 45532800, 52141320, 179734464, 301953024, 311323824
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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 3-smooth number. Of the 937 harmonic numbers below 10^14, 38 are terms in this sequence.
If a term is not a harmonic number, then its numerator and denominator of the harmonic mean of its divisors are powers of 2 and 3, or vice versa.
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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Amiram Eldar, Table of n, a(n) for n = 1..47
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EXAMPLE
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2 is a term since the harmonic mean of its divisors is 4/3 = 2^2/3.
3 is a term since the harmonic mean of its divisors is 3/2.
40 is a term since the harmonic mean of its divisors is 32/9 = 2^5/3^2.
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MATHEMATICA
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smQ[n_] := n == 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := smQ[Numerator[(hn = h[n])]] && smQ[Denominator[hn]]; Select[Range[10^5], q]
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CROSSREFS
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Cf. A000079, A000244, A001599, A001600, A003586, A099377, A099378.
Subsequence of A348868.
Similar sequences: A074266, A122254, A348658, A348659.
Sequence in context: A269996 A336240 A336458 * A018318 A277809 A051717
Adjacent sequences: A348864 A348865 A348866 * A348868 A348869 A348870
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KEYWORD
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nonn
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AUTHOR
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Amiram Eldar, Nov 02 2021
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STATUS
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approved
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