

A335270


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


2




OFFSET

1,1


COMMENTS

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.
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


LINKS

Table of n, a(n) for n=1..6.


EXAMPLE

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.


MATHEMATICA

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10^5], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega1), usigma[#]  1] &]


CROSSREFS

The unitary version of A247077.
Cf. A006086, A000961, A024619, A034444, A034448, A077610, A309307, A335268, A335269.
Sequence in context: A252220 A263301 A053174 * A103837 A302755 A064245
Adjacent sequences: A335267 A335268 A335269 * A335271 A335272 A335273


KEYWORD

nonn,hard,more


AUTHOR

Amiram Eldar, May 29 2020


EXTENSIONS

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


STATUS

approved



