A335269 -id:A335269 - cp's OEIS frontend

A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.

6, 15, 20, 24, 28, 30, 45, 60, 72, 90, 91, 96, 100, 112, 153, 216, 220, 240, 264, 272, 325, 352, 360, 364, 378, 496, 703, 765, 780, 816, 832, 1056, 1125, 1170, 1225, 1360, 1431, 1512, 1656, 1760, 1891, 1900, 1984, 2275, 2448, 2520, 2701, 2912, 3024, 3168, 3321

Offset: 1

Amiram Eldar, May 29 2020

nonn

Since the unitary divisors of a power of prime (A000961), p^e, are {1, p^e}, they are trivial terms and hence they are excluded from this sequence.
The corresponding harmonic means are 3, 5, 6, 6, 7, 5, 9, 7, 12, 7, 13, 8, 10, 14, 17, ...
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m) | m * (2^omega(m)-1), or A063919(m) | (m * A309307(m)), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) = A034444(m) is the number of the unitary divisors of m.
The squarefree terms of A335267 are also terms of this sequence.
The terms with 2 distinct prime divisors are of the form p^e * (2*p^e - 1), when the second factor is also a prime power. The least term which both of its 2 prime divisors are nonunitary (with multiplicity larger than 1) is 1225 = 5^2 * 7^2 = 5^2 * (2 * 5^2 - 1).
The unitary perfect numbers (A002827) are terms of this sequence: if m is a unitary perfect number then usigma(m)-m = m.

			6 is a term since its unitary divisors other than 1 are 2, 3 and 6, and their harmonic mean, 3/(1/2 + 1/3 + 1/6) = 3, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..1000

The unitary version of A335267.
A002827 is subsequence.
Cf. A006086, A000961, A024619, A034444, A034448, A063919, A077610, A309307, A335269, A335270.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3000], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - #] &]

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

A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.

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