A335269 Numbers for which the harmonic mean of the nontrivial unitary divisors is an integer.

228, 345, 1645, 2120, 4025, 4386, 4977, 7725, 8041, 13026, 23881, 24157, 24336, 51925, 88473, 115957, 150161, 169893, 229177, 255041, 278721, 322592, 342637, 377201, 490725, 538625, 656937, 1497517, 1566981, 2132021, 3256261, 3847001, 4646101, 5054221, 5524897

Offset: 1

Author

Amiram Eldar, May 29 2020

Keywords

• nonn

Comments

A number m is a term if the set {d|m ; d > 1, d < m, gcd(d, m/d) = 1} is nonempty and the harmonic mean its members is an integer.

The corresponding harmonic means are 8, 9, 15, 16, 25, 12, 21, 15, 33, 12, ...

Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m-1) | m*(2^omega(m)-2), where usigma is the sum of unitary divisors (A034448), and 2^omega(m)-2 = A034444(m)-2 = A087893 (m) is the number of the nontrivial unitary divisors of m.

The squarefree terms of A247078 are also terms of this sequence.

Examples

			228 is a term since the harmonic mean of its nontrivial unitary divisors, {3, 4, 12, 19, 57, 76} is 8 which is an integer.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..150

Crossrefs

The unitary version of A247078.

Cf. A006086, A034444, A034448, A077610, A087893, A335268, A335270.

Programs

Mathematica

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10^6], (omega = PrimeNu[#]) > 1 && Divisible[#*(2^omega - 2), usigma[#] - # - 1] &]