A247077 Composite numbers for which the harmonic mean of proper divisors is an integer.

1645, 88473, 63626653506

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..3.

Example

The proper divisors of 1645 are [1,5,7,35,47,235,329] and their harmonic mean is 7/(1/1 + 1/5 + 1/7 + 1/35 + 1/47 + 1/235 + 1/329) = 5.

Mathematica

Select[Range[2, 100000], (IntegerQ[HarmonicMean[Most[Divisors[#]]]] && Not[PrimeQ[#]])&] (* Daniel Lignon, Nov 17 2014 *)

Prog

(PARI) lista(nn) = forcomposite (n=2, nn, my(d=divisors(n)); if (denominator((#d-1)/sum(i=1, #d-1, 1/d[i])) == 1, print1(n, ", "))); \\ Michel Marcus, Nov 17 2014
(PARI) isok(n) = n > 1 && !isprime(n) && (n*(numdiv(n)-1)) % (sigma(n)-1) == 0; \\ Daniel Suteu, Feb 05 2021

Crossrefs

Cf. A001599 for harmonic mean of all divisors and A247078 for harmonic mean of nontrivial divisors.

Sequence in context: A135016 A238066 A349178 * A093059 A255620 A253030

Adjacent sequences: A247074 A247075 A247076 * A247078 A247079 A247080

Keyword

nonn,more,bref

Author

Daniel Lignon, Nov 17 2014

Extensions

a(3) from Hiroaki Yamanouchi, Nov 20 2014

Status

approved