A348965 Exponential harmonic numbers of type 2 that are not squarefree.

12, 18, 36, 40, 60, 75, 84, 90, 120, 126, 132, 135, 150, 156, 180, 198, 204, 208, 228, 234, 252, 270, 276, 280, 306, 342, 348, 360, 372, 396, 414, 420, 440, 444, 450, 468, 492, 516, 520, 522, 525, 540, 544, 558, 564, 588, 600, 612, 624, 630, 636, 660, 666, 675

Offset: 1

Author

Amiram Eldar, Nov 05 2021

Keywords

• nonn

Comments

Sándor (2006) proved that all squarefree numbers are exponential harmonic numbers of type 2.

Examples

			12 = 2^2 * 3 is a term since it is not squarefree, its exponential divisors are 6 and 12, and their harmonic mean, 8, is an integer.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
• József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

Crossrefs

Intersection of A013929 and A348964.

Cf. A005117, A348962.

Programs

Mathematica

f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^(e-#) &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[1000], ! SquareFreeQ[#] && ehQ[#] &]