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 A348867 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 3-smooth numbers. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS Amiram Eldar, Table of n, a(n) for n = 1..47 EXAMPLE 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. MATHEMATICA 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] CROSSREFS 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 KEYWORD nonn AUTHOR Amiram Eldar, Nov 02 2021 STATUS approved

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Last modified May 20 23:05 EDT 2022. Contains 353886 sequences. (Running on oeis4.)