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 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 (list; graph; refs; listen; history; text; internal format)
 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 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. 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

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Last modified November 29 18:41 EST 2021. Contains 349416 sequences. (Running on oeis4.)