A335268 -id:A335268 - cp's OEIS frontend

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

Amiram Eldar, May 29 2020

nonn

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.

			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.

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

The unitary version of A247078.

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

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

A335270 Numbers that are not powers of primes (A024619) whose harmonic mean of their proper unitary divisors is an integer.

Original entry on oeis.org

228, 1645, 7725, 88473, 20295895122, 22550994580

Offset: 1

Amiram Eldar, May 29 2020

Since 1 is the only proper unitary divisor of powers of prime (A000961), they are trivial terms and hence they are excluded from this sequence.
The corresponding harmonic means are 4, 5, 5, 9, 18, 20.
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-1) | m*(2^omega(m)-1), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) - 1 = A034444(m) - 1 = A309307(m) is the number of the proper unitary divisors of m.
The squarefree terms of A247077 are also terms of this sequence.
a(7) > 10^12, if it exists. - Giovanni Resta, May 30 2020
Conjecture: all terms are of the form n*(usigma(n)-1) where usigma(n)-1 is prime. - Chai Wah Wu, Dec 17 2020

			228 is a term since the harmonic mean of its proper unitary divisors, {1, 3, 4, 12, 19, 57, 76} is 4 which is an integer.

The unitary version of A247077.

Cf. A006086, A000961, A024619, A034444, A034448, A077610, A309307, A335268, A335269.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10^5], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - 1] &]

a(5)-a(6) from Giovanni Resta, May 30 2020

cp's OEIS Frontend

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

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