cp's OEIS Frontend

Of course, for all prime numbers the harmonic mean of proper divisors is an integer.

a(4) >= 2*10^11. - Hiroaki Yamanouchi, Nov 20 2014

Conjecture: all terms are of the form m*(sigma(m)-1) where sigma(m)-1 is prime. - Chai Wah Wu, Dec 15 2020

a(4) <= 22351741783447265625. - Daniel Suteu, Dec 16 2020

From Chai Wah Wu, Feb 04 2021: (Start)

Other terms of the sequence of the form m*(sigma(m)-1) correspond to the following values of m:

3 * 5^143

3 * 5^623

3 * 5^1423

5 * 7^127

5 * 7^6595

101 * 103^25

(End)

Equivalently, composite numbers k such that sigma(k)-1 divides k*(tau(k)-1), where sigma = A000203 and tau = A000005. - Daniel Suteu, Feb 05 2021

The primes are excluded from this sequence since they are trivial terms.

The corresponding harmonic means are 3, 5, 5, 5, 13, 9, 9, 9, 9, 9, 37, ...

Equivalently, composite numbers m such that (sigma(m)-m) | m*(tau(m)-1), or A001065(m) | A168014(m).

The semiprimes terms of this sequence are of the form p*q where p and q = 2*p - 1 are primes (A129521).

If m is a k-perfect numbers, k = 2, 3, ... (i.e., sigma(m) = k*m), then sigma(m)-m = (k-1)*m. If (k-1)*m | m*(tau(m)-1) then (k-1) | (tau(m)-1). If k is odd then tau(m) is also odd, so m is a square, and sigma(m) is odd. Since m | sigma(m) this means that m is also odd. Since there is no known odd multiply-perfect number except for 1 (A007691), there are no known k-perfect numbers with odd k in this sequence.

The perfect numbers (k=2, A000396) are terms: if m is a perfect number then sigma(m)-m = m.

The 4-perfect number (k=4, A027687) m are terms if 3 | (tau(m)-1). Of the first 36 terms of A027687 there are 8 such terms, the first is A027687(26).

The 6-perfect number (k=6, A046061) m are terms if 5 | (tau(m)-1). Of the first 245 terms of A046061 there are 20 such terms, the first is A046061(19).

Hemiperfect numbers that are terms of this sequence include A055153(i) for i = 10, 18 and 20, A141645(21), and A159271(i) for i = 97 and 103.

A number m is a term if the set {d|m ; d > 1, d < m, gcd(d, m/d) = 1} is nonempty and the harmonic mean its members is an integer.

The corresponding harmonic means are 8, 9, 15, 16, 25, 12, 21, 15, 33, 12, ...

Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m-1) | m*(2^omega(m)-2), where usigma is the sum of unitary divisors (A034448), and 2^omega(m)-2 = A034444(m)-2 = A087893 (m) is the number of the nontrivial unitary divisors of m.

The squarefree terms of A247078 are also terms of this sequence.

That's the numbers which are in A247078 and not in A001248.

a(149) >= 2*10^11. - Hiroaki Yamanouchi, Nov 20 2014