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

1,1

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.

Links

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

Example

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.

Mathematica

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

Crossrefs

The unitary version of A247078.

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

Sequence in context: A252451 A305063 A258550 * A252228 A252221 A154519

Adjacent sequences: A335266 A335267 A335268 * A335270 A335271 A335272

Keyword

nonn

Author

Amiram Eldar, May 29 2020

Status

approved