A348867 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 3-smooth numbers.

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

Offset: 1

Author

Amiram Eldar, Nov 02 2021

Keywords

• nonn

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.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..47

Crossrefs

Cf. A000079, A000244, A001599, A001600, A003586, A099377, A099378.

Subsequence of A348868.

Similar sequences: A074266, A122254, A348658, A348659.

Programs

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]