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

228, 1645, 7725, 88473, 20295895122, 22550994580

Offset: 1

Author

Amiram Eldar, May 29 2020

Keywords

• nonn
• hard
• more

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

Examples

			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.
		

Crossrefs

The unitary version of A247077.

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

Programs

Mathematica

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

Extensions

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