login
A335270
Numbers that are not powers of primes (A024619) whose harmonic mean of their proper unitary divisors is an integer.
2
228, 1645, 7725, 88473, 20295895122, 22550994580
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, 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
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^omega-1), usigma[#] - 1] &]
CROSSREFS
The unitary version of A247077.
Sequence in context: A252220 A263301 A053174 * A103837 A302755 A064245
KEYWORD
nonn,hard,more
AUTHOR
Amiram Eldar, May 29 2020
EXTENSIONS
a(5)-a(6) from Giovanni Resta, May 30 2020
STATUS
approved