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A247077 Composite numbers for which the harmonic mean of proper divisors is an integer. 3
1645, 88473, 63626653506 (list; graph; refs; listen; history; text; internal format)
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

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